2.1 Kummer surfaces of product type, the Inose fibration and the Inose surface

Let $E_{1}$ and $E_{2}$ be two elliptic curves over $k$ . Later in this paper, we will be concerned with fields of definition of the Mordell–Weil groups of various elliptic fibrations. Here, we give a summary of Kummer surfaces and related constructions, being careful about the field of definition.

Let $\mathit{Km}(E_{1}\times E_{2})$ be the Kummer surface associated with the product abelian surface $E_{1}\times E_{2}$ , namely the minimal desingularization of the quotient surface $E_{1}\times E_{2}/\{\pm 1\}$ . If $E_{1}$ and $E_{2}$ are defined by the equations

(2.1) $$\begin{eqnarray}\displaystyle \begin{array}{@{}l@{}}E_{1}:y^{2}=x^{3}+ax+b,\\ E_{2}:y^{2}=x^{3}+cx+d,\end{array} & & \displaystyle\end{eqnarray}$$

an affine singular model of $\mathit{Km}(E_{1}\times E_{2})$ may be given as the hypersurface in $\mathbb{A}^{3}$ defined by the equation

(2.2) $$\begin{eqnarray}x_{2}^{3}+cx_{2}+d=t_{2}^{2}(x_{1}^{3}+ax_{1}+b).\end{eqnarray}$$

Then the map $\mathit{Km}(E_{1}\times E_{2})\rightarrow \mathbb{P}^{1}$ induced by $(x_{1},x_{2},t_{2})\mapsto t_{2}$ is an elliptic fibration, which is sometimes called the Kummer pencil. This elliptic fibration has obvious geometric sections (i.e., sections defined over $\bar{k}$ ), but they are defined only over the extension $k(E_{1}[2],E_{2}[2])/k$ obtained by adjoining the coordinates of points of order $2$ .

Take a parameter $t_{6}$ such that $t_{2}=t_{6}^{3}$ , and consider (2.2) as a family of cubic curves in $\mathbb{P}^{2}$ over the field $k(t_{6})$ . Then, this family has a rational point $(1:t_{6}^{2}:0)$ (cf. Mestre [Reference MestreMe] and Kuwata–Wang [Reference Kuwata and WangKwW]). Using this point, we convert (2.2) to the Weierstrass form:

(2.3) $$\begin{eqnarray}Y^{2}=X^{3}-3\,ac\,X+\frac{1}{64}\Bigl(\unicode[STIX]{x1D6E5}_{E_{1}}t_{6}^{6}+864\,bd+\frac{\unicode[STIX]{x1D6E5}_{E_{2}}}{t_{6}^{6}}\Bigr),\end{eqnarray}$$

where $\unicode[STIX]{x1D6E5}_{E_{1}}$ and $\unicode[STIX]{x1D6E5}_{E_{2}}$ are the discriminants of $E_{1}$ and $E_{2}$ , respectively:

$$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{E_{1}}=-16(4a^{3}+27b^{2}),\qquad \unicode[STIX]{x1D6E5}_{E_{2}}=-16(4c^{3}+27d^{2}).\end{eqnarray}$$

The change of coordinates between (2.2) and (2.3) are given by

(2.4) $$\begin{eqnarray}\displaystyle \left\{\begin{array}{@{}l@{}}\displaystyle X=\frac{-t_{6}^{2}(2at_{6}^{4}-c)x_{1}-3(bt_{6}^{6}-d)-(at_{6}^{4}-2c)x_{2}}{t_{6}^{2}(t_{6}^{2}x_{1}-x_{2})},\\ \displaystyle Y=\frac{6(at_{6}^{4}-c)(bt_{6}^{6}-d)+6(at_{6}^{4}-c)(at_{6}^{6}x_{1}-cx_{2})-9(bt_{6}^{6}-d)(t_{6}^{4}x_{1}^{2}-x_{2}^{2})}{2t_{6}^{3}(t_{6}^{2}x_{1}-x_{2})^{2}}.\end{array}\right. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Note that if we choose other models of $E_{1}$ and $E_{2}$ , we still obtain an isomorphic equation. Indeed, if we replace the equations of $E_{1}$ and $E_{2}$ by

$$\begin{eqnarray}\displaystyle & & \displaystyle E_{1}:y^{2}=x^{3}+(k^{4}a)x+(k^{6}b),\nonumber\\ \displaystyle & & \displaystyle E_{2}:y^{2}=x^{3}+(l^{4}c)x+(l^{6}d),\nonumber\end{eqnarray}$$

then replacing $(X,Y,t_{6})$ by $(l^{4}X,l^{6}Y,(l/k)t_{6})$ , we recover equation (2.3).

It is easy to see that equation (2.3) is invariant under the two automorphisms of the $t_{6}$ -line:

(2.5) $$\begin{eqnarray}\begin{array}{@{}l@{}}\unicode[STIX]{x1D70E}:t_{6}\mapsto \unicode[STIX]{x1D701}_{6}t_{6},\quad \text{where}~\unicode[STIX]{x1D701}_{6}~\text{is a primitive sixth root of unity},\\ \unicode[STIX]{x1D70F}:t_{6}\mapsto \unicode[STIX]{x1D6FF}/t_{6},\quad \text{where}~\unicode[STIX]{x1D6FF}~\text{is a chosen sixth root of}~\unicode[STIX]{x1D6E5}_{2}/\unicode[STIX]{x1D6E5}_{1}.\end{array}\end{eqnarray}$$

Taking the quotient by the action of $\unicode[STIX]{x1D70E}$ , or equivalently, setting $t_{1}=t_{6}^{6}$ , we obtain an elliptic curve over the field $k(t_{1})$ , which we denote by $F_{E_{1},E_{2}}^{(1)}$ :

(2.6) $$\begin{eqnarray}F_{E_{1},E_{2}}^{(1)}:Y^{2}=X^{3}-3\,ac\,X+\frac{1}{64}\Bigl(\unicode[STIX]{x1D6E5}_{E_{1}}t_{1}+864\,bd+\frac{\unicode[STIX]{x1D6E5}_{E_{2}}}{t_{1}}\Bigr).\end{eqnarray}$$

Definition 2.3. For $n=1,\ldots ,6$ , let $t_{n}$ be a parameter satisfying $t_{n}^{n}=t_{1}$ . Define the elliptic curve $F_{E_{1},E_{2}}^{(n)}$ over $k(t_{n})$ by

$$\begin{eqnarray}F_{E_{1},E_{2}}^{(n)}:Y^{2}=X^{3}-3\,ac\,X+\frac{1}{64}\Bigl(\unicode[STIX]{x1D6E5}_{E_{1}}t_{n}^{n}+864\,bd+\frac{\unicode[STIX]{x1D6E5}_{E_{2}}}{t_{n}^{n}}\Bigr).\end{eqnarray}$$

When $E_{1}$ and $E_{2}$ are understood, we write $F^{(n)}$ for short.

By Inose’s theorem [Reference InoseI1, Cor. 1.2], the Picard number of the K3 surface $F^{(n)}$ does not depend on  $n$ , and equals the Picard number of the Kummer surface $\mathit{Km}(E_{1}\times E_{2})$ . It is therefore at least  $18$ . These surfaces are clearly of geometric and arithmetic interest, being closely related to abelian surfaces which are the product of two elliptic curves. We now summarize what is known about the geometric Picard and Mordell–Weil groups of these elliptic K3 surfaces.

Define $R(t)$ and $S(t)$ by letting the Inose surface as in equation (2.6) be $Y^{2}=X^{3}+R(t)X+S(t)$ , and let $h=\operatorname{rank}\operatorname{Hom}(E_{1},E_{2})$ , so that $0\leqslant h\leqslant 4$ . The table below list the minimal Weierstrass equations, the configuration of singular fibers, and the Mordell–Weil rank in the “generic” case $j(E_{1})\neq j(E_{2})$ and $j(E_{i})\neq 0$ . In the other cases, which will not be relevant to this paper, we refer the reader to [Reference KuwataKw2, Th. 4.1] for the analogous data.

The Néron–Severi and transcendental lattices were further analyzed by Shioda [Reference ShiodaSh3, Reference ShiodaSh5, Reference ShiodaSh7], culminating in the following theorems, which are stated in the geometric situation $k=\mathbb{C}$ . In this case we may scale $x,y,t$ to work with a simpler equation of $F^{(n)}$ , as in [Reference InoseI1, Reference ShiodaSh3]:

$$\begin{eqnarray}Y^{2}=X^{3}-3\sqrt[3]{J_{1}J_{2}}\,X+t^{n}+\frac{1}{t^{n}}-2\sqrt{(1-J_{1})(1-J_{2})},\end{eqnarray}$$

where $J_{i}=j(E_{i})/1728$ .

Theorem 2.7. (Shioda [Reference ShiodaSh7])

There is a natural isomorphism of lattices

$$\begin{eqnarray}\operatorname{Hom}(E_{1},E_{2})\cong F^{(1)}(k(t)).\end{eqnarray}$$

In particular, we can compute the Mordell–Weil rank as follows:

Proposition 2.8. [Reference ShiodaSh3]

For elliptic curves $E_{1}$ and $E_{2}$ , and $1\leqslant n\leqslant 6$ , we have

$$\begin{eqnarray}\displaystyle \operatorname{rank}F_{E_{1},E_{2}}^{(n)}(\bar{k}(t_{n})) & = & \displaystyle h+\min (4(n-1),16)\nonumber\\ \displaystyle & & \displaystyle -\,\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~j(E_{1})\neq j(E_{2}),\\ n\quad & \text{if}~j(E_{1})=j(E_{2})\neq 0,1728,\\ 2n\quad & \text{if}~j(E_{1})=j(E_{2})=0~\text{or}~1728,\end{array}\right.\nonumber\end{eqnarray}$$

where $h=\operatorname{rank}\operatorname{Hom}(E_{1},E_{2})$ .

In particular, the largest possible Mordell–Weil rank is $18$ , and we have the following.

Shioda further analyzed the surface $F^{(5)}$ for the CM elliptic curves $y^{2}=x^{3}-1$ and $y^{2}=x^{3}-15x+22$ , which are $2$ -isogenous to each other, and determined its Mordell–Weil group [Reference ShiodaSh6]. For the same pair of elliptic curves, $F^{(6)}$ was studied in [Reference Chahal, Meijer and TopCMT] and generators for its Mordell–Weil group were computed.

In this article, we will generalize these results further, to obtain explicit descriptions of the Mordell–Weil lattices of the surfaces $F^{(n)}$ . Our main results are the following.

2.2 Galois correspondence of sublattices

The Mordell–Weil lattice of the surface $F^{(6)}$ has a particularly rich structure, with sublattices induced from the Mordell–Weil lattices of several quotients which are elliptic rational or K3 surfaces. As we saw in (2.5), a dihedral group $D_{6}$ generated by $\unicode[STIX]{x1D70E}$ and $\unicode[STIX]{x1D70F}$ acts on $F^{(6)}$ . We define

(2.7) $$\begin{eqnarray}s_{n,i}=t_{n}+\frac{(\unicode[STIX]{x1D701}_{6}^{-i}\unicode[STIX]{x1D6FF})^{6/n}}{t_{n}},\quad n=1,2,3,6,\;i=0,1,\ldots ,n-1.\end{eqnarray}$$

Recall that $t_{n}^{n}=t_{1}=t_{6}^{6}$ . Then $s_{n,i}$ is invariant under $\unicode[STIX]{x1D70E}^{n}$ and $\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70E}^{i}$ . Write $S=s_{1,0}=t_{1}+(\unicode[STIX]{x1D6E5}_{E_{2}}/\unicode[STIX]{x1D6E5}_{E_{1}})t_{1}^{-1}$ for simplicity. Then, the extension $k(t_{6})/k(S)$ is a Galois extension, and its Galois group is $D_{6}=\langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\rangle$ . Our basic idea is to consider the elliptic surface

$$\begin{eqnarray}F_{S}:Y^{2}=X^{3}-{\textstyle \frac{1}{3}}A\,X+{\textstyle \frac{1}{64}}(\unicode[STIX]{x1D6E5}_{E_{1}}S+C),\end{eqnarray}$$

where $A$ and $C$ are as in (2.8), and view $F^{(6)}(\bar{k}(t_{6}))$ as the Mordell–Weil group of $F_{S}$ over the extension $\bar{k}(t_{6})/\bar{k}(S)$ . In other words, we regard $F^{(6)}(\bar{k}(t_{6}))=F_{S}(\bar{k}(t_{6}))$ .

Write $M(t_{n})=F_{S}(\bar{k}(t_{n}))$ and $M(s_{n,i})=F_{S}(\bar{k}(s_{n,i}))$ . Between $k(t_{6})$ and $k(S)$ there are fourteen intermediate fields. Corresponding to these we have a relation among the Mordell–Weil groups $M(t_{n})$ and $M(s_{n,i})$ .

For later use, let us write a more general formula for $F^{(n)}$ . If $E_{1}$ and $E_{2}$ are given by

$$\begin{eqnarray}\displaystyle E_{1}:y^{2} & = & \displaystyle x^{3}+a_{2}x^{2}+a_{4}x+a_{6},\nonumber\\ \displaystyle E_{2}:y^{2} & = & \displaystyle x^{3}+a_{2}^{\prime }x^{2}+a_{4}^{\prime }x+a_{6}^{\prime },\nonumber\end{eqnarray}$$

then the equation of $F_{E_{1},E_{2}}^{(n)}$ is given by

$$\begin{eqnarray}F_{E_{1},E_{2}}^{(n)}:Y^{2}=X^{3}-\frac{1}{3}A\,X+\frac{1}{64}\left(B\,t_{n}^{n}+C+\frac{D}{t_{n}^{n}}\right),\end{eqnarray}$$

where

(2.8) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}A=(a_{2}^{2}-3a_{4})({a_{2}^{\prime }}^{2}-3a_{4}^{\prime }),\\ B=16(a_{2}^{2}a_{4}^{2}-4a_{2}^{3}a_{6}+18a_{2}a_{4}a_{6}-4a_{4}^{3}-27a_{6}^{2})=\unicode[STIX]{x1D6E5}_{E_{1}},\\ C=\frac{32}{27}(2a_{2}^{3}-9a_{2}a_{4}+27a_{6})(2{a_{2}^{\prime }}^{3}-9a_{2}^{\prime }a_{4}^{\prime }+27a_{6}^{\prime }),\\ D=16({a_{2}^{\prime }}^{2}{a_{4}^{\prime }}^{2}-4{a_{2}^{\prime }}^{3}a_{6}^{\prime }+18a_{2}^{\prime }a_{4}^{\prime }a_{6}^{\prime }-4{a_{4}^{\prime }}^{3}-27{a_{6}^{\prime }}^{2})=\unicode[STIX]{x1D6E5}_{E_{2}}.\end{array}\right.\end{eqnarray}$$

3.1 $F^{(1)}$ for isogenous case

The Mordell–Weil lattice of $F^{(1)}(\overline{k}(t_{1}))$ for generic $E_{1}$ and $E_{2}$ is trivial, and we have nothing to do. If $E_{1}$ and $E_{2}$ are isogenous but not isomorphic over $\overline{k}$ , we have the following interpretation of the Mordell–Weil lattice by $\operatorname{Hom}(E_{1},E_{2})$ .

Proposition 3.1. (Shioda [Reference ShiodaSh5])

Let $E_{1}$ and $E_{2}$ be two elliptic curves not isomorphic to each other over  $\overline{k}$ . Then, the Mordell–Weil lattice of $F^{(1)}(\overline{k}(t_{1}))$ is isomorphic to the lattice $\operatorname{Hom}(E_{1},E_{2})\langle 2\rangle$ , where the pairing of $\operatorname{Hom}(E_{1},E_{2})$ is given by

$$\begin{eqnarray}(\unicode[STIX]{x1D711},\unicode[STIX]{x1D713})={\textstyle \frac{1}{2}}(\deg (\unicode[STIX]{x1D711}+\unicode[STIX]{x1D713})-\deg \unicode[STIX]{x1D711}-\deg \unicode[STIX]{x1D713})\quad \unicode[STIX]{x1D711},\unicode[STIX]{x1D713}\in \operatorname{Hom}(E_{1},E_{2}).\end{eqnarray}$$

For a given $\unicode[STIX]{x1D711}\in \operatorname{Hom}(E_{1},E_{2})$ , we would like to compute the section corresponding to $\unicode[STIX]{x1D711}$ explicitly. To do so, we consider the inclusion

$$\begin{eqnarray}\operatorname{Hom}(E_{1},E_{2})\langle 2\rangle \simeq F_{E_{1},E_{2}}^{(1)}(\bar{k}(t_{1})){\hookrightarrow}\operatorname{Hom}(E_{1},E_{2})\langle 12\rangle \subset F_{E_{1},E_{2}}^{(6)}(\bar{k}(t_{6}))\end{eqnarray}$$

induced by $t_{1}\mapsto t_{6}^{6}$ , and we look for a section in $F_{E_{1},E_{2}}^{(6)}(\bar{k}(t_{6}))$ .

Suppose $E_{1}$ and $E_{2}$ are given in the form of (2.1). By replacing $t_{2}$ by $t_{6}^{3}$ in the equation of Kummer surface (2.2), we regard it as a cubic curve over $k(t_{6})$ . More precisely, we consider the cubic curve in the projective plane over $k(t_{6})$ with coordinates $(x_{1}:x_{2}:z)$ defined by

(3.1) $$\begin{eqnarray}C_{t_{6}}:x_{2}^{3}+cx_{2}z^{2}+dz^{3}=t_{6}^{6}(x_{1}^{3}+ax_{1}z^{2}+bz^{3}).\end{eqnarray}$$

Suppose $\unicode[STIX]{x1D711}$ is an isogeny of degree  $d$ . Then, $\unicode[STIX]{x1D711}$ can be written in the form

$$\begin{eqnarray}\unicode[STIX]{x1D711}:(x_{1},y_{1})\mapsto (x_{2},y_{2})=(\unicode[STIX]{x1D711}_{x}(x_{1}),\unicode[STIX]{x1D711}_{y}(x_{1})y_{1}).\end{eqnarray}$$

Consider the curve of degree $d$ given by $x_{2}=\unicode[STIX]{x1D711}_{x}(x_{1})$ . The intersection of these two curves

(3.2) $$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}x_{2}^{3}+cx_{2}+d=t_{6}^{6}(x_{1}^{3}+ax_{1}+b),\\ x_{2}=\unicode[STIX]{x1D711}_{x}(x_{1}),\end{array}\right.\end{eqnarray}$$

gives a divisor of degree $3d$ in $C_{t_{6}}$ . Since we have $\unicode[STIX]{x1D711}_{x}(x_{1})^{3}+c\unicode[STIX]{x1D711}_{x}(x_{1})+d=\unicode[STIX]{x1D711}_{y}(x_{1})^{2}y_{1}^{2}=\unicode[STIX]{x1D711}_{y}(x_{1})^{2}(x_{1}^{3}+ax_{1}+b),$ the first equation reduces to

(3.3) $$\begin{eqnarray}(\unicode[STIX]{x1D711}_{y}(x_{1})-t_{6}^{3})(\unicode[STIX]{x1D711}_{y}(x_{1})+t_{6}^{3})(x_{1}^{3}+ax_{1}+b)=0.\end{eqnarray}$$

To compute $P_{\unicode[STIX]{x1D711}}^{\pm }$ explicitly, we need to find a curve in the plane passing through the points in the divisor $D_{\unicode[STIX]{x1D711}}^{\pm }$ , and this is in principle just an exercise in linear algebra. We shall illustrate it using a concrete example (see Example 9.2).

3.2 $F^{(2)}$ for generic case

Suppose two elliptic curves are given by

(3.4) $$\begin{eqnarray}\begin{array}{@{}l@{}}E_{\unicode[STIX]{x1D706}}:y^{2}=x(x-1)(x-\unicode[STIX]{x1D706}),\\ E_{\unicode[STIX]{x1D707}}:y^{2}=x(x-1)(x-\unicode[STIX]{x1D707})\end{array}\end{eqnarray}$$

for $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \overline{k}$ . Then, $F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(2)}$ is given by the Weierstrass equation

(3.5) $$\begin{eqnarray}\displaystyle F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(2)}:Y^{2} & = & \displaystyle X^{3}-\frac{1}{3}(\unicode[STIX]{x1D707}^{2}-\unicode[STIX]{x1D707}+1)(\unicode[STIX]{x1D706}^{2}-\unicode[STIX]{x1D706}+1)X\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{4}\unicode[STIX]{x1D706}^{2}(\unicode[STIX]{x1D706}-1)^{2}t^{2}+\frac{\unicode[STIX]{x1D707}^{2}(\unicode[STIX]{x1D707}-1)^{2}}{4t^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{54}(2\unicode[STIX]{x1D707}-1)(\unicode[STIX]{x1D707}-2)(\unicode[STIX]{x1D707}+1)(2\unicode[STIX]{x1D706}-1)(\unicode[STIX]{x1D706}-2)(\unicode[STIX]{x1D706}+1).\end{eqnarray}$$

Here, we wrote $t_{2}=t$ for simplicity. Note that in this case the equation of the Kummer surface $x_{2}(x_{2}-1)(x_{2}-\unicode[STIX]{x1D707})=t_{2}^{2}x_{1}(x_{1}-1)(x_{1}-\unicode[STIX]{x1D706})$ can be converted over $k(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$ to the Weierstrass form $F^{(2)}$ using the point $(x_{1},x_{2})=(0,0)$ . Then we can obtain sections from $2$ -torsion points of $E_{\unicode[STIX]{x1D706}}$ and $E_{\unicode[STIX]{x1D707}}$ . In the following proposition, $P_{1},\ldots ,P_{4}$ are obtained from $(x_{1},x_{2})=(1,1),(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),(1,\unicode[STIX]{x1D707}),(\unicode[STIX]{x1D706},1)$ , respectively.

Proposition 3.3. Suppose $E_{\unicode[STIX]{x1D706}}$ and $E_{\unicode[STIX]{x1D707}}$ are not isogenous over $\bar{k}$ . Then the following sections form a basis of the Mordell–Weil group $F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(2)}(\bar{k}(t))$

$$\begin{eqnarray}\displaystyle P_{1} & = & \displaystyle \bigg(\frac{1}{3}(-2\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}+1),\frac{1}{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)t+\frac{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}-1)}{2t}\bigg)\nonumber\\ \displaystyle P_{2} & = & \displaystyle \bigg(\frac{1}{3}(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-2),\frac{1}{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)t+\frac{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}-1)}{2t}\bigg)\nonumber\\ \displaystyle P_{3} & = & \displaystyle \bigg(\frac{1}{3}(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-2\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}+1),\frac{1}{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)t-\frac{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}-1)}{2t}\bigg)\nonumber\\ \displaystyle P_{4} & = & \displaystyle \bigg(\frac{1}{3}(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D706}-2\unicode[STIX]{x1D707}+1),\frac{1}{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)t-\frac{\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}-1)}{2t}\bigg).\nonumber\end{eqnarray}$$

Moreover, the height pairing of these sections is given by

$$\begin{eqnarray}\frac{2}{3}\left(\begin{array}{@{}rrrr@{}}2 & -1 & 0 & 0\\ -1 & 2 & 0 & 0\\ 0 & 0 & 2 & -1\\ 0 & 0 & -1 & 2\end{array}\right).\end{eqnarray}$$

As a lattice $F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(2)}(\bar{k}(t))\simeq A_{2}^{\ast }\langle 2\rangle \oplus A_{2}^{\ast }\langle 2\rangle$ .

If $E_{1}$ and $E_{2}$ are isogenous but not isomorphic, the sublattice $\operatorname{Hom}(E_{1},E_{2})\langle 4\rangle \oplus A_{2}^{\ast }\langle 2\rangle \oplus A_{2}^{\ast }\langle 2\rangle$ of the Mordell–Weil lattice has index $2^{h}$ , where $h=\operatorname{rank}\operatorname{Hom}(E_{1},E_{2})$ (see [Reference ShiodaSh5, Theorem 1.2]).

4.1 Elliptic modular surface associated with $\unicode[STIX]{x1D6E4}(3)$

Let us begin with the Hesse cubic $x^{3}+y^{3}+z^{3}=3\unicode[STIX]{x1D707}xyz$ . Using the point $(-1:1:0)$ , we convert it to the Weierstrass form (see [Reference Rubin and SilverbergRS1]):

(4.1) $$\begin{eqnarray}y^{2}=x^{3}-27\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}^{3}+8)x+54(\unicode[STIX]{x1D707}^{6}-20\unicode[STIX]{x1D707}^{3}-8).\end{eqnarray}$$

This elliptic curve has nine $3$ -torsion points that are defined over $k(\unicode[STIX]{x1D714})$ , where $\unicode[STIX]{x1D714}$ is a primitive cube root of unity.

The group of $3$ -torsion points is generated by

$$\begin{eqnarray}\displaystyle T_{1} & = & \displaystyle (3(\unicode[STIX]{x1D707}+2)^{2},36(\unicode[STIX]{x1D707}^{2}+\unicode[STIX]{x1D707}+1))\qquad \text{and}\qquad \nonumber\\ \displaystyle T_{2} & = & \displaystyle (-9\unicode[STIX]{x1D707}^{2},12(2\unicode[STIX]{x1D714}+1)(\unicode[STIX]{x1D707}^{2}+\unicode[STIX]{x1D707}+1)).\nonumber\end{eqnarray}$$

Let

$$\begin{eqnarray}\unicode[STIX]{x1D703}=\left(\begin{array}{@{}cc@{}}1 & 1\\ 0 & 1\end{array}\right)\qquad \text{and}\qquad \unicode[STIX]{x1D70C}=\left(\begin{array}{@{}rc@{}}0 & 1\\ -1 & 0\end{array}\right)\end{eqnarray}$$

be generators for $G=\operatorname{SL}_{2}(\mathbb{F}_{3})$ . Consider the following representation $\unicode[STIX]{x1D70B}$ of $G$ on the projective line $\mathbb{P}_{\unicode[STIX]{x1D707}}^{1}$ by fractional linear transformations, which factors through $\operatorname{PSL}_{2}(\mathbb{F}_{3})\cong A_{4}$ .

$$\begin{eqnarray}\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}):\unicode[STIX]{x1D707}\mapsto \unicode[STIX]{x1D714}\unicode[STIX]{x1D707},\qquad \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D70C}):\unicode[STIX]{x1D707}\mapsto \frac{\unicode[STIX]{x1D707}+2}{\unicode[STIX]{x1D707}-1}.\end{eqnarray}$$

The $j$ -invariant of (4.1) is given by

$$\begin{eqnarray}j=\frac{27\unicode[STIX]{x1D707}^{3}(\unicode[STIX]{x1D707}+8)^{3}}{(\unicode[STIX]{x1D707}^{3}-1)^{3}},\end{eqnarray}$$

and is invariant under the action of $\unicode[STIX]{x1D70B}(\operatorname{SL}_{2}(\mathbb{F}_{3}))$ .

Now we regard (4.1) as an elliptic surface. In other words, consider the elliptic modular surface ${\mathcal{E}}_{\unicode[STIX]{x1D6E4}(3)}\rightarrow X(3)\simeq \mathbb{P}_{\unicode[STIX]{x1D707}}^{1}$ whose generic fiber at $\unicode[STIX]{x1D707}$ is given by (4.1).

Lemma 4.1. The action of $G=\operatorname{SL}_{2}(\mathbb{F}_{3})$ on the base $\mathbb{P}_{\unicode[STIX]{x1D707}}^{1}$ extends to a compatible faithful action $\widetilde{\unicode[STIX]{x1D70B}}$ on the surface ${\mathcal{E}}_{\unicode[STIX]{x1D6E4}(3)}$ . It is given by the formulas

(4.2) $$\begin{eqnarray}\displaystyle \begin{array}{@{}c@{}}\displaystyle \widetilde{\unicode[STIX]{x1D70B}}(\unicode[STIX]{x1D703}):(x,y,\unicode[STIX]{x1D707})\mapsto (\unicode[STIX]{x1D714}^{2}x,y,\unicode[STIX]{x1D714}\unicode[STIX]{x1D707}),\\ \displaystyle \widetilde{\unicode[STIX]{x1D70B}}(\unicode[STIX]{x1D70C}):(x,y,\unicode[STIX]{x1D707})\mapsto \left(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(\unicode[STIX]{x1D707}-1)^{2}}x,\frac{(2\unicode[STIX]{x1D714}+1)^{3}}{(\unicode[STIX]{x1D707}-1)^{3}}y,\frac{\unicode[STIX]{x1D707}+2}{\unicode[STIX]{x1D707}-1}\right).\end{array} & & \displaystyle\end{eqnarray}$$

The action $\widetilde{\unicode[STIX]{x1D70B}}$ in turn induces an action $\unicode[STIX]{x1D6F1}$ of $G$ on the group of sections ${\mathcal{E}}_{\unicode[STIX]{x1D6E4}(3)}(k(\unicode[STIX]{x1D707}))$ of the elliptic surface as follows. For a section $s:\mathbb{P}_{\unicode[STIX]{x1D707}}^{1}\rightarrow {\mathcal{E}}_{\unicode[STIX]{x1D6E4}(3)}$ given by rational functions $\unicode[STIX]{x1D707}\mapsto P(\unicode[STIX]{x1D707})=(x(\unicode[STIX]{x1D707}),y(\unicode[STIX]{x1D707}))$ , $\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6FE})(s)$ for $\unicode[STIX]{x1D6FE}\in G$ is defined to be $\unicode[STIX]{x1D707}\mapsto P^{\prime }(\unicode[STIX]{x1D707})$ where

$$\begin{eqnarray}P^{\prime }(\unicode[STIX]{x1D707})=\widetilde{\unicode[STIX]{x1D70B}}(\unicode[STIX]{x1D6FE})(P(\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6FE}^{-1})(\unicode[STIX]{x1D707}))).\end{eqnarray}$$

The action of $G$ by $\unicode[STIX]{x1D6F1}$ on the subgroup of $3$ -torsion sections ${\mathcal{E}}_{\unicode[STIX]{x1D6E4}(3)}(k(\unicode[STIX]{x1D707}))[3]$ is equivalent to the usual linear action of $\operatorname{SL}_{2}(\mathbb{F}_{3})$ on $\mathbb{F}_{3}^{2}$ (and identifies $G$ as the automorphism group of the $3$ -torsion subgroup equipped with the Weil pairing). More precisely, we have

$$\begin{eqnarray}\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D703}):\left\{\begin{array}{@{}l@{}}T_{1}\mapsto T_{1}-T_{2}\\ T_{2}\mapsto T_{2},\end{array}\right.\qquad \unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D70C}):\left\{\begin{array}{@{}l@{}}T_{1}\mapsto T_{2}\\ T_{2}\mapsto -T_{1}.\end{array}\right.\end{eqnarray}$$

4.2 $F^{(3)}$ for universal families

Now, take two copies of (4.1),

$$\begin{eqnarray}\displaystyle E_{u}:y_{1}^{2} & = & \displaystyle x_{1}^{3}-27u(u^{3}+8)x_{1}+54(u^{6}-20u^{3}-8),\nonumber\\ \displaystyle E_{v}:y_{2}^{2} & = & \displaystyle x_{2}^{3}-27v(v^{3}+8)x_{2}+54(v^{6}-20v^{3}-8),\nonumber\end{eqnarray}$$

and construct $\mathit{Ino}(E_{u},E_{v})$ , and in turn, $F_{E_{u},E_{v}}^{(3)}$ :

(4.3) $$\begin{eqnarray}\displaystyle F_{E_{u},E_{v}}^{(3)}:Y^{2} & = & \displaystyle X^{3}-27uv(u^{3}+8)(v^{3}+8)X\nonumber\\ \displaystyle & & \displaystyle +\,1728(u^{3}-1)^{3}t_{3}^{3}+\frac{1728(v^{3}-1)^{3}}{t_{3}^{3}}\nonumber\\ \displaystyle & & \displaystyle +\,54(u^{6}-20u^{3}-8)(v^{6}-20v^{3}-8).\end{eqnarray}$$

Here, since $\unicode[STIX]{x1D6E5}_{E_{1}}=2^{12}\cdot 3^{9}(\unicode[STIX]{x1D707}^{3}-1)^{3}$ , and so forth, we scaled $X$ and $Y$ differently from (2.8); the difference is a Weierstrass transformation over  $\mathbb{Q}$ .

Let ${\mathcal{E}}_{u}\rightarrow \mathbb{P}_{u}^{1}$ and ${\mathcal{E}}_{v}\rightarrow \mathbb{P}_{v}^{1}$ be the elliptic modular surfaces associated with $E_{u}$ and $E_{v}$ , respectively. Also let $G_{u}=\langle \unicode[STIX]{x1D703}_{u},\unicode[STIX]{x1D70C}_{u}\rangle$ and $G_{v}=\langle \unicode[STIX]{x1D703}_{v},\unicode[STIX]{x1D70C}_{v}\rangle$ be groups of automorphisms of ${\mathcal{E}}_{u}$ and ${\mathcal{E}}_{v}$ described above, respectively. We consider (4.3) as the family of elliptic surfaces ${\mathcal{F}}_{E_{u},E_{v}}^{(3)}\rightarrow \mathbb{P}_{u}^{1}\times \mathbb{P}_{v}^{1}$ parametrized by $u$ and $v$ . The total space is a fourfold.

Proposition 4.3. The actions of $G_{u}$ and $G_{v}$ induce the action $\widetilde{\unicode[STIX]{x1D6F1}}$ on the fourfold ${\mathcal{F}}_{E_{u},E_{v}}^{(3)}$ given by the following formulas.

$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D703}_{u}):(X,Y,t_{3},u,v)\mapsto (\unicode[STIX]{x1D714}^{2}X,Y,\unicode[STIX]{x1D714}t_{3},\unicode[STIX]{x1D714}u,v) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D703}_{v}):(X,Y,t_{3},u,v)\mapsto (\unicode[STIX]{x1D714}^{2}X,Y,\unicode[STIX]{x1D714}^{2}t_{3},u,\unicode[STIX]{x1D714}v), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D70C}_{u}):(X,Y,t_{3},u,v)\mapsto \left(\!\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(u-1)^{2}}X,\frac{(2\unicode[STIX]{x1D714}+1)^{3}}{(u-1)^{3}}Y,\frac{(u-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}t_{3},\frac{u+2}{u-1},v\!\right)\!, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D70C}_{v}):(X,Y,t_{3},u,v)\mapsto \left(\!\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(v-1)^{2}}X,\frac{(2\unicode[STIX]{x1D714}+1)^{3}}{(v-1)^{3}}Y,\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(v-1)^{2}}t_{3},u,\frac{v+2}{v-1}\!\right)\!. & \displaystyle \nonumber\end{eqnarray}$$

Proof. Recall that $F^{(6)}$ is, by definition, birationally equivalent to

$$\begin{eqnarray}x_{2}^{3}+cx_{2}+d=t_{6}^{6}(x_{1}^{3}+ax_{1}+b),\end{eqnarray}$$

where $a,b,c,d$ are appropriate functions of $u,v$ . (Note also that $F^{(3)}$ is not birationally equivalent to $x_{2}^{3}+cx_{2}+d=t_{3}^{3}(x_{1}^{3}+ax_{1}+b)$ , the latter being a rational surface.) The action on ${\mathcal{E}}_{u}$ or ${\mathcal{E}}_{v}$ induces the action on $(X,Y,t_{6})$ through the change of variables (2.4). The change of variables also depends on the choice of the rational point $(1:t_{6}^{2}:0)$ .

Since $\unicode[STIX]{x1D703}_{u}$ acts as $(x_{1},y_{1},u)\mapsto (\unicode[STIX]{x1D714}^{2}x_{1},y_{1},\unicode[STIX]{x1D714}u)$ , it stabilizes the equation $x_{2}^{3}+cx_{2}+d=t_{6}^{6}(x_{1}^{3}+ax_{1}+b)$ , but it moves the rational point $(1:t_{6}^{2}:0)$ to $(\unicode[STIX]{x1D714}^{2}:t_{6}^{2}:0)=\big(1:(\unicode[STIX]{x1D714}^{2}t_{6})^{2}:0\big)$ . Therefore, $t_{6}\mapsto \unicode[STIX]{x1D714}^{2}t_{6}$ under this transformation. Using (2.4), we see that $X$ is mapped to $\unicode[STIX]{x1D714}^{2}X$ and $Y$ remains invariant. Thus, $\unicode[STIX]{x1D703}_{u}$ acts as $(X,Y,t_{6},u,v)\mapsto (\unicode[STIX]{x1D714}^{2}X,Y,\unicode[STIX]{x1D714}^{2}t_{6},\unicode[STIX]{x1D714}u,v)$ on ${\mathcal{F}}^{(6)}$ . This action descends to the above expression for $\widetilde{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D703}_{u})$ on ${\mathcal{F}}_{E_{u},E_{v}}^{(3)}$ .

As for $\unicode[STIX]{x1D70C}_{u}$ , the action

$$\begin{eqnarray}\displaystyle (x_{1},x_{2},t_{6},u,v)\mapsto \left(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(u-1)^{2}}x_{1},x_{2},\frac{(u-1)}{(2\unicode[STIX]{x1D714}+1)}t_{6},\frac{(u+2)}{(u-1)},v\right) & & \displaystyle \nonumber\end{eqnarray}$$

stabilizes the equation, and by following the change of coordinates (2.4), we obtain the desired formula. Similarly, we obtain the expressions for $\unicode[STIX]{x1D703}_{v}$ and  $\unicode[STIX]{x1D70C}_{v}$ .◻

For clarity of notation, we will henceforth use just $\unicode[STIX]{x1D6FE}$ instead of $\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D6FE}),\widetilde{\unicode[STIX]{x1D70B}}(\unicode[STIX]{x1D6FE}),\unicode[STIX]{x1D6F1}(\unicode[STIX]{x1D6FE}),\widetilde{\unicode[STIX]{x1D6F1}}(\unicode[STIX]{x1D6FE})$ , with the action being understood.

Corollary 4.4. The group $G_{u}\times G_{v}$ acts on the group of sections $F_{E_{u},E_{v}}^{(3)}(\bar{k}(u,v)(t_{3}))$ . More precisely, if $\unicode[STIX]{x1D6FE}\in G_{u}\times G_{v}$ and $s\in F_{E_{u},E_{v}}^{(3)}(\bar{k}(u,v)(t_{3}))$ is given by $X(t_{3},u,v)$ and $Y(t_{3},u,v)$ , then $\unicode[STIX]{x1D6FE}\cdot s$ is given by

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}X\big(\unicode[STIX]{x1D6FE}^{-1}t_{3},\unicode[STIX]{x1D6FE}^{-1}u,\unicode[STIX]{x1D6FE}^{-1}v\big)\qquad \text{and}\qquad \unicode[STIX]{x1D6FE}Y\big(\unicode[STIX]{x1D6FE}^{-1}t_{3},\unicode[STIX]{x1D6FE}^{-1}u,\unicode[STIX]{x1D6FE}^{-1}v\big).\end{eqnarray}$$

From the action of $D_{6}$ on $F^{(6)}$ (see (2.5)), we see that $\langle \unicode[STIX]{x1D70E}^{2},\unicode[STIX]{x1D70F}\rangle \simeq D_{3}\simeq S_{3}$ acts on $F^{(3)}$ . We summarize the action on a section $(X(t_{3},u,v),Y(t_{3},u,v))$ as follows.

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}^{2}\big(X(t_{3},u,v),Y(t_{3},u,v)\big)=\big(X(\unicode[STIX]{x1D714}^{2}t_{3},u,v),Y(\unicode[STIX]{x1D714}^{2}t_{3},u,v)\big) & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D70F}\big(X(t_{3},u,v),Y(t_{3},u,v)\big)=\big(X(\unicode[STIX]{x1D6FF}^{2}/t_{3},u,v),Y(\unicode[STIX]{x1D6FF}^{2}/t_{3},u,v)\big) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D703}_{u}\big(X(t_{3},u,v),Y(t_{3},u,v)\big)=\big(\unicode[STIX]{x1D714}^{2}X(\unicode[STIX]{x1D714}^{2}t_{3},\unicode[STIX]{x1D714}^{2}u,v),Y(\unicode[STIX]{x1D714}^{2}t_{3},\unicode[STIX]{x1D714}^{2}u,v)\big) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D703}_{v}\big(X(t_{3},u,v),Y(t_{3},u,v)\big)=\big(\unicode[STIX]{x1D714}^{2}X(\unicode[STIX]{x1D714}t_{3},u,\unicode[STIX]{x1D714}^{2}v),Y(\unicode[STIX]{x1D714}t_{3},u,\unicode[STIX]{x1D714}^{2}v)\big) & \displaystyle \nonumber\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{u}\big(X(t_{3},u,v),Y(t_{3},u,v)\big) & = & \displaystyle \left(\frac{(u-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}X\Bigl(\frac{(u-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}t_{3},\frac{u+2}{u-1},v\Bigr),\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,\frac{(u-1)^{3}}{(2\unicode[STIX]{x1D714}+1)^{3}}Y\Bigl(\frac{(u-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}t_{3},\frac{u+2}{u-1},v\Bigr)\right)\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{v}\big(X(t_{3},u,v),Y(t_{3},u,v)\big) & = & \displaystyle \left(\frac{(v-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}X\Bigl(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(v-1)^{2}}t_{3},u,\frac{v+2}{v-1}\Bigr),\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,\frac{(v-1)^{3}}{(2\unicode[STIX]{x1D714}+1)^{3}}Y\Bigl(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(v-1)^{2}}t_{3},u,\frac{v+2}{v-1}\Bigr)\right).\nonumber\end{eqnarray}$$

We may expect that the orbit of a section by these automorphisms generates the whole Mordell–Weil group, and it turns out that it is the case. To show this we must find one section, and we will do this in the following subsections.

4.3 Computation of $M(s_{3,0})$

As we modified the equation of $F^{(3)}$ from the one in Section 2, we also change the definition of $s_{3,i}$ to

$$\begin{eqnarray}s_{3,i}=4\Bigl((u^{3}-1)t_{3}+\frac{v^{3}-1}{\unicode[STIX]{x1D714}^{i}t_{3}}\Bigr).\end{eqnarray}$$

In what follows we denote $s_{3,0}$ simply by $s$ . Then, (4.3) becomes

(4.4) $$\begin{eqnarray}\displaystyle R_{s}^{(3)}:Y^{2} & = & \displaystyle X^{3}-27uv(u^{3}+8)(v^{3}+8)X\nonumber\\ \displaystyle & & \displaystyle +\,27(s^{3}-48(u^{3}-1)(v^{3}-1)s\nonumber\\ \displaystyle & & \displaystyle +\,2(u^{6}-20u^{3}-8)(v^{6}-20v^{3}-8)).\end{eqnarray}$$

$R_{s}^{(3)}$ is a rational elliptic surface over $k(u,v)$ that has a singular fiber of type I $_{0}^{\ast }$ at $s=\infty$ . As a consequence, its Mordell–Weil lattice is of type $D_{4}^{\ast }$ (type $\text{n}^{\circ }~9$ in the Oguiso–Shioda [Reference Oguiso and ShiodaOS] classification). The Mordell–Weil lattice of a rational elliptic surface is generated by the section of the form $(c_{2}s^{2}+c_{1}s+c_{0},d_{3}s^{3}+d^{2}s^{2}+d_{1}s+d_{0})$ . An easy calculation shows that the fact that our elliptic surface has a singular fiber of type I $_{0}^{\ast }$ at $s=\infty$ implies $c_{2}=d_{3}=d_{2}=0$ . Some further calculations show that we have a point $P_{0}$ given by

$$\begin{eqnarray}\displaystyle P_{0} & = & \displaystyle \Bigl(-3s-9(uv+2)^{2},\nonumber\\ \displaystyle & & \displaystyle \quad (2\unicode[STIX]{x1D714}+1)^{3}(3(uv+2)s+4(u^{3}-1)(v^{3}-1)+36(u^{2}v^{2}+uv+1))\Bigr).\nonumber\end{eqnarray}$$

The actions of $\unicode[STIX]{x1D703}_{u}$ and $\unicode[STIX]{x1D703}_{v}$ do not leave $M(s_{3,0})=R_{s}^{(3)}(\overline{k}(s))$ invariant, but $\bar{\unicode[STIX]{x1D703}}_{u}:=\unicode[STIX]{x1D70E}^{4}\circ \unicode[STIX]{x1D703}_{u}$ and $\bar{\unicode[STIX]{x1D703}}_{v}:=\unicode[STIX]{x1D70E}^{2}\circ \unicode[STIX]{x1D703}_{v}$ do. In fact, it is easily verified that the action of $\bar{\unicode[STIX]{x1D703}}_{u}$ and $\bar{\unicode[STIX]{x1D703}}_{v}$ on $F^{(3)}$ fix $s$ . On the other hand, $\unicode[STIX]{x1D70C}_{u}$ and $\unicode[STIX]{x1D70C}_{v}$ leave $M(s_{3,0})$ invariant; they take $s$ to $(2\unicode[STIX]{x1D714}+1)^{2}/(u-1)^{2}s$ and $(2\unicode[STIX]{x1D714}+1)^{2}/(v-1)^{2}$ , respectively. The actions of $\bar{\unicode[STIX]{x1D703}}_{u}$ , $\bar{\unicode[STIX]{x1D703}}_{v}$ , $\unicode[STIX]{x1D70C}_{u}$ and $\unicode[STIX]{x1D70C}_{v}$ on $M(s_{3,0})$ are given as follows.

$$\begin{eqnarray}\displaystyle \bar{\unicode[STIX]{x1D703}}_{u}\big(X(s,u,v),Y(s,u,v)\big) & = & \displaystyle \big(\unicode[STIX]{x1D714}^{2}X(s,\unicode[STIX]{x1D714}^{2}u,v),Y(s,\unicode[STIX]{x1D714}^{2}u,v)\big)\nonumber\\ \displaystyle \bar{\unicode[STIX]{x1D703}}_{v}\big(X(s,u,v),Y(s,u,v)\big) & = & \displaystyle \big(\unicode[STIX]{x1D714}^{2}X(s,u,\unicode[STIX]{x1D714}^{2}v),Y(s,u,\unicode[STIX]{x1D714}^{2}v)\big)\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{u}\big(X(s,u,v),Y(s,u,v)\big) & = & \displaystyle \left(\frac{(u-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}X\Bigl(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(u-1)^{2}}s,\frac{u+2}{u-1},v\Bigr),\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.-\,\frac{(u-1)^{3}}{(2\unicode[STIX]{x1D714}+1)^{3}}Y\Bigl(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(u-1)^{2}}s,\frac{u+2}{u-1},v\Bigr)\right)\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{v}\big(X(s,u,v),Y(s,u,v)\big) & = & \displaystyle \left(\frac{(v-1)^{2}}{(2\unicode[STIX]{x1D714}+1)^{2}}X\Bigl(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(v-1)^{2}}s,u,\frac{v+2}{v-1}\Bigr),\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.-\,\frac{(v-1)^{3}}{(2\unicode[STIX]{x1D714}+1)^{3}}Y\Bigl(\frac{(2\unicode[STIX]{x1D714}+1)^{2}}{(v-1)^{2}}s,u,\frac{v+2}{v-1}\Bigr)\right).\nonumber\end{eqnarray}$$

Proposition 4.5. The group of sections

$$\begin{eqnarray}M(s_{3,0})=R_{s}^{(3)}(\bar{k}(s))\subset F_{E_{u},E_{v}}^{(3)}(\bar{k}(t_{3}))\end{eqnarray}$$

is invariant under the automorphisms $\bar{\unicode[STIX]{x1D703}}_{u}$ , $\bar{\unicode[STIX]{x1D703}}_{v}$ , $\unicode[STIX]{x1D70C}_{u}$ , $\unicode[STIX]{x1D70C}_{v}$ . The group $\langle \bar{\unicode[STIX]{x1D703}}_{u},\bar{\unicode[STIX]{x1D703}}_{v},\unicode[STIX]{x1D70C}_{u},\unicode[STIX]{x1D70C}_{v}\rangle$ acts on the lattice $M(s_{3,0})$ , and $M(s_{3,0})$ is generated by

$$\begin{eqnarray}P_{0},\qquad \bar{\unicode[STIX]{x1D703}}_{u}(P_{0}),\qquad (\bar{\unicode[STIX]{x1D703}}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0}),\qquad \text{and}\qquad (\unicode[STIX]{x1D70C}_{u}^{-1}\bar{\unicode[STIX]{x1D703}}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0}).\end{eqnarray}$$

Furthermore, the height pairing matrix with respect to these sections is given by

$$\begin{eqnarray}\frac{1}{2}\left(\begin{array}{@{}rrrr@{}}2 & -1 & -1 & -1\\ -1 & 2 & 0 & 0\\ -1 & 0 & 2 & 0\\ -1 & 0 & 0 & 2\end{array}\right).\end{eqnarray}$$

All the twenty-four sections of minimal height $1$ are obtained from $P_{0}$ by the automorphism group $\langle \bar{\unicode[STIX]{x1D703}}_{u},\unicode[STIX]{x1D70C}_{u}\rangle$ .

Remark 4.6. The $X$ -coordinates of four sections above are as follows.

$$\begin{eqnarray}\displaystyle X(P_{0}) & = & \displaystyle -3s-9(uv+2)^{2},\nonumber\\ \displaystyle X(\bar{\unicode[STIX]{x1D703}}_{u}(P_{0})) & = & \displaystyle -3\unicode[STIX]{x1D714}^{2}s-9(uv+2\unicode[STIX]{x1D714})^{2},\nonumber\\ \displaystyle X((\bar{\unicode[STIX]{x1D703}}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0})) & = & \displaystyle -3\unicode[STIX]{x1D714}^{2}s+3(uv+2u+2\unicode[STIX]{x1D714}v-2\unicode[STIX]{x1D714})^{2},\nonumber\\ \displaystyle X((\unicode[STIX]{x1D70C}_{u}^{-1}\bar{\unicode[STIX]{x1D703}}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0})) & = & \displaystyle -3\unicode[STIX]{x1D714}^{2}s+3(uv+2\unicode[STIX]{x1D714}^{2}u+2\unicode[STIX]{x1D714}^{2}v-2\unicode[STIX]{x1D714})^{2}.\nonumber\end{eqnarray}$$

The $Y$ -coordinates are a little more complicated and we omit them here.Footnote ²

4.4 $F^{(3)}(\overline{k}(t_{3}))$ in the generic case

If we consider $E_{u}$ and $E_{v}$ as universal curves with independent variables $u$ and $v$ , they are not isogenous curves. Looking at the diagram at the end of Section 2, $F^{(3)}(\bar{k}(t_{3}))$ contains four sublattices, $F^{(1)}(\bar{k}(t_{1}))$ , $M(s_{3,0})$ , $M(s_{3,1})$ and $M(s_{3,2})$ . In our case $F^{(1)}(\bar{k}(t_{1}))$ is trivial.

Proof. From the identity

$$\begin{eqnarray}\displaystyle s^{3}-3s & = & \displaystyle \Bigl(t+\frac{1}{t}\Bigr)^{3}-3\Bigl(t+\frac{1}{t}\Bigr)=t^{3}+\frac{1}{t^{3}}\nonumber\\ \displaystyle & = & \displaystyle t^{3}+\frac{1}{(\unicode[STIX]{x1D714}t)^{3}}=\Bigl(t+\frac{1}{\unicode[STIX]{x1D714}t}\Bigr)^{3}-3\unicode[STIX]{x1D714}^{2}\Bigl(t+\frac{1}{\unicode[STIX]{x1D714}t}\Bigr)=s_{1}^{3}-3\unicode[STIX]{x1D714}^{2}s_{1},\nonumber\end{eqnarray}$$

we see that the lattice $M(s_{3,1})$ is the Mordell–Weil lattice of the elliptic curve given by the equation obtained by replacing $s$ by $\unicode[STIX]{x1D714}^{2}s_{3,1}$ in (4.4). The assertion is now clear from

$$\begin{eqnarray}s_{3,1}=4\unicode[STIX]{x1D714}\left((u^{3}-1)\unicode[STIX]{x1D714}^{2}t_{3}+\frac{(v^{3}-1)}{\unicode[STIX]{x1D714}^{2}t_{3}}\right)=\unicode[STIX]{x1D714}\unicode[STIX]{x1D70E}^{2}(s_{3,0}).\end{eqnarray}$$

Similarly for $M(s_{3,2})$ .◻

Theorem 4.8. Let $u,v\in \overline{k}$ be such that $E_{u}$ and $E_{v}$ are not isogenous. The Mordell–Weil group $F_{E_{u},E_{v}}^{(3)}(\bar{k}(t_{3}))$ for the elliptic curve $F_{E_{u},E_{v}}^{(3)}$ over $k(u,v)$ defined by (4.3) are generated by $M(s_{3,0})$ and $M(s_{3,1})$ . As a lattice, $F_{E_{u},E_{v}}^{(3)}(\bar{k}(t_{3}))$ is generated by

$$\begin{eqnarray}\displaystyle & \displaystyle (\unicode[STIX]{x1D70C}_{u}^{-1}\unicode[STIX]{x1D703}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0}),\qquad (\unicode[STIX]{x1D703}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0}),\qquad \unicode[STIX]{x1D703}_{u}(P_{0}),\qquad P_{0}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D70E}^{2}(P_{0}),\qquad (\unicode[STIX]{x1D70E}^{2}\unicode[STIX]{x1D703}_{u})(P_{0}),\qquad (\unicode[STIX]{x1D70E}^{2}\unicode[STIX]{x1D703}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0}),\qquad \text{and}\qquad (\unicode[STIX]{x1D70E}^{2}\unicode[STIX]{x1D70C}_{u}^{-1}\unicode[STIX]{x1D703}_{u}\unicode[STIX]{x1D70C}_{u})(P_{0}) & \displaystyle \nonumber\end{eqnarray}$$

with the height pairing matrix

$$\begin{eqnarray}\frac{1}{2}\left(\begin{array}{@{}rrrrrrrr@{}}4 & 0 & 0 & 1 & -2 & 0 & 0 & -2\\ 0 & 4 & 0 & 1 & -2 & 0 & -2 & 0\\ 0 & 0 & 4 & 1 & -2 & -2 & 0 & 0\\ 1 & 1 & 1 & 4 & -2 & 1 & 1 & 1\\ -2 & -2 & -2 & -2 & 4 & 1 & 1 & 1\\ 0 & 0 & -2 & 1 & 1 & 4 & 0 & 0\\ 0 & -2 & 0 & 1 & 1 & 0 & 4 & 0\\ -2 & 0 & 0 & 1 & 1 & 0 & 0 & 4\end{array}\right).\end{eqnarray}$$

5.1 Elliptic modular surface associated with $\unicode[STIX]{x1D6E4}(4)$

The elliptic modular surface over the modular curve $X(4)$ is given by

(5.1) $$\begin{eqnarray}E_{\unicode[STIX]{x1D70E}}:y^{2}=x(x+(\unicode[STIX]{x1D70E}^{2}+1)^{2})(x+(\unicode[STIX]{x1D70E}^{2}-1)^{2})\end{eqnarray}$$

(cf. [Reference ShiodaSh1]). The subgroup of $4$ -torsion points are generated by

$$\begin{eqnarray}(-\unicode[STIX]{x1D70E}^{4}+1,2\unicode[STIX]{x1D70E}^{4}-2)\qquad \text{and}\qquad (-(\unicode[STIX]{x1D70E}^{2}-1)(\unicode[STIX]{x1D70E}+i)^{2},-2\unicode[STIX]{x1D70E}(\unicode[STIX]{x1D70E}^{2}-1)(\unicode[STIX]{x1D70E}+i)^{2}),\end{eqnarray}$$

where $i=\sqrt{-1}$ . The $j$ -invariant of (5.1) is given by

$$\begin{eqnarray}j=\frac{16(\unicode[STIX]{x1D70E}^{8}+14\unicode[STIX]{x1D70E}^{4}+1)^{3}}{\unicode[STIX]{x1D70E}^{4}(\unicode[STIX]{x1D70E}^{4}-1)^{4}}.\end{eqnarray}$$

Let

$$\begin{eqnarray}\unicode[STIX]{x1D703}=\left(\begin{array}{@{}cc@{}}1 & 1\\ 0 & 1\end{array}\right)\qquad \text{and}\qquad \unicode[STIX]{x1D70C}=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right)\end{eqnarray}$$

be generators for $G=\operatorname{SL}_{2}(\mathbb{Z}/4\mathbb{Z})$ . Consider the following representation $\unicode[STIX]{x1D70B}$ of $G$ on the projective line $\mathbb{P}_{\unicode[STIX]{x1D70E}}^{1}$ by fractional linear transformations, which factors through $\operatorname{PSL}_{2}(\mathbb{Z}/4\mathbb{Z})$ .

$$\begin{eqnarray}\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}):\unicode[STIX]{x1D70E}\mapsto i\unicode[STIX]{x1D70E},\qquad \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D70C}):\unicode[STIX]{x1D70E}\mapsto \frac{\unicode[STIX]{x1D70E}-2}{\unicode[STIX]{x1D70E}+1}.\end{eqnarray}$$

The $j$ -invariant is invariant under the action of $\unicode[STIX]{x1D70B}(\operatorname{SL}_{2}(\mathbb{Z}/4\mathbb{Z}))$ .

We take two copies of the elliptic modular surface (5.1):

$$\begin{eqnarray}\displaystyle E_{u}:y_{1}^{2} & = & \displaystyle x_{1}(x_{1}+(u^{2}+1)^{2})(x_{1}+(u^{2}-1)^{2}),\nonumber\\ \displaystyle E_{v}:y_{2}^{2} & = & \displaystyle x_{2}(x_{2}+(v^{2}+1)^{2})(x_{2}+(v^{2}-1)^{2}).\nonumber\end{eqnarray}$$

We then obtain $F_{E_{u},E_{v}}^{(4)}$ :

(5.2) $$\begin{eqnarray}\displaystyle F_{E_{u},E_{v}}^{(4)}:Y^{2} & = & \displaystyle X^{3}-27(u^{8}+14u^{4}+1)(v^{8}+14v^{4}+1)X\nonumber\\ \displaystyle & & \displaystyle +\,54\Bigl(54u^{4}(u^{4}-1)^{4}t^{4}+\frac{54v^{4}(v^{4}-1)^{4}}{t^{4}}\nonumber\\ \displaystyle & & \displaystyle +\,(u^{12}-33u^{8}-33u^{4}+1)(v^{12}-33v^{8}-33v^{4}+1)\Bigr).\end{eqnarray}$$

5.2 Rational elliptic surfaces

As before we set

$$\begin{eqnarray}s_{j}=i^{j}u(u^{4}-1)t+\frac{v(v^{4}-1)}{t}.\end{eqnarray}$$

where $i=\sqrt{-1}$ . Then we may write the equation of $F^{(4)}$ as

$$\begin{eqnarray}R_{s_{j}}^{(4)}:Y^{2}=X^{3}-AX+4(s_{j}^{4}-4i^{j}Bs_{j}^{2}+2i^{2j}B^{2})+C,\end{eqnarray}$$

where

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}A=27(u^{8}+14u^{4}+1)(v^{8}+14v^{4}+1)\\ B=uv(u^{4}-1)(v^{4}-1)\\ C=54(u^{12}-33u^{8}-33u^{4}+1)(v^{12}-33v^{8}-33v^{4}+1).\end{array}\right.\end{eqnarray}$$

This is a rational elliptic surface over $\mathbb{P}_{s_{j}}^{1}$ , with a $\text{IV}$ fiber over $s_{j}=\infty$ . Generically, this is the only reducible fiber, and the Mordell–Weil lattice is therefore $E_{6}^{\ast }$ . To describe the sections, note that it suffices to do so for $R_{s_{0}}^{(4)}$ , since $\big(X(u,v,s_{0}),Y(u,v,s_{0})\big)$ is a section of $R_{s_{0}}^{(4)}$ if and only if $\big(X(i^{j}u,v,s_{i}),Y(i^{j}u,v,s_{i})\big)$ is a section of $R_{s_{j}}^{(4)}$ .

There are fifty-four sections of $R_{s_{0}}^{(4)}$ of minimal height, with twenty-seven sections intersecting each non-identity component of the $\text{IV}$ fiber.

To solve for the sections intersecting one of these, we set

$$\begin{eqnarray}X=x_{0}+x_{1}s,\qquad Y=y_{0}+y_{1}s+54s^{2},\end{eqnarray}$$

(where we have written $s=s_{0}$ for ease of notation) and substitute into the Weierstrass equation. It is then easy to solve for the remaining coefficients. We obtain a basis of sections; we list the $x$ -coordinates here for brevity. (The full sections may be found in the auxiliary files).

$$\begin{eqnarray}\displaystyle x(P_{1}) & = & \displaystyle -18(1+i)(uv+i)s\nonumber\\ \displaystyle & & \displaystyle -\,3(5u^{4}v^{4}-u^{4}-v^{4}+24iu^{3}v^{3}-24u^{2}v^{2}-24iuv+5),\nonumber\\ \displaystyle x(P_{2}) & = & \displaystyle 18(1+i)(uv+i)s\nonumber\\ \displaystyle & & \displaystyle -\,3(5u^{4}v^{4}-u^{4}-v^{4}+24iu^{3}v^{3}-24u^{2}v^{2}-24iuv+5),\nonumber\\ \displaystyle x(P_{3}) & = & \displaystyle 18(1-i)(uv-i)s\nonumber\\ \displaystyle & & \displaystyle -\,3(5u^{4}v^{4}-u^{4}-v^{4}-24iu^{3}v^{3}-24u^{2}v^{2}+24iuv+5),\nonumber\\ \displaystyle x(P_{4}) & = & \displaystyle 18(1-i)(u+iv)s\nonumber\\ \displaystyle & & \displaystyle +\,3(u^{4}v^{4}-5u^{4}-5v^{4}-24iu^{3}v+24u^{2}v^{2}+24iuv^{3}+1),\nonumber\\ \displaystyle x(P_{5}) & = & \displaystyle -18(1+i)(u-iv)s\nonumber\\ \displaystyle & & \displaystyle +\,3(u^{4}v^{4}-5u^{4}-5v^{4}+24iu^{3}v+24u^{2}v^{2}-24iuv^{3}+1),\nonumber\\ \displaystyle x(P_{6}) & = & \displaystyle 3(u^{4}v^{4}+u^{4}+v^{4}+6u^{4}v^{2}+6u^{2}v^{4}-12u^{2}v^{2}+6u^{2}+6v^{2}+1).\nonumber\end{eqnarray}$$

5.3 The Mordell–Weil group in the generic case

Let $P_{1},\ldots ,P_{6}$ be the sections obtained from $R_{s_{0}}^{(4)}$ , and $P_{1}^{\prime },\ldots ,P_{6}^{\prime }$ the corresponding sections from $R_{s_{1}}^{(4)}$ (obtained by substituting $iu$ for $u$ in $P_{i}$ ). Together, they do not quite span the whole Mordell–Weil lattice of $F^{(4)}$ . We define

$$\begin{eqnarray}\displaystyle Q_{1} & = & \displaystyle -(P_{1}-P_{3}+P_{4}-P_{5}+P_{6})/2,\nonumber\\ \displaystyle Q_{1}^{\prime } & = & \displaystyle -(P_{1}^{\prime }-P_{3}^{\prime }+P_{4}^{\prime }-P_{5}^{\prime }+P_{6}^{\prime })/2;\nonumber\end{eqnarray}$$

expressions for these sections may be obtained from the computer files. Note that they are well defined by the equation above, since $\operatorname{MW}(F^{(4)})$ is torsion-free by Theorem 2.5.

5.4 $F^{(4)}$ as a quartic surface

The minimal nonsingular model of $F_{E_{u},E_{v}}^{(4)}$ is isomorphic to the quartic surface defined by

(5.3) $$\begin{eqnarray}\displaystyle & & \displaystyle S_{4}:ZW(Z+(v^{2}+1)^{2}W)(Z+(v^{2}-1)^{2}W)\nonumber\\ \displaystyle & & \displaystyle \qquad =XY(X+(u^{2}+1)^{2}Y)(X+(u^{2}-1)^{2}Y).\end{eqnarray}$$

$F_{E_{u},E_{v}}^{(4)}$ corresponds to the elliptic fibration on $S_{4}$ defined by the elliptic parameter $t_{4}=Y/W$ . Generically, this quartic surface contains sixteen lines (cf. [Reference SegreSe, Reference InoseI1, Reference KuwataKw1]). They are obtained as the intersection of one of the four planes

$$\begin{eqnarray}X=0,\qquad Y=0,\qquad X+(u^{2}+1)^{2}Y=0,\qquad X+(u^{2}-1)^{2}Y=0\end{eqnarray}$$

and one of the four planes

$$\begin{eqnarray}Z=0,\qquad W=0,\qquad Z+(v^{2}+1)^{2}W=0,\qquad Z+(v^{2}-1)^{2}W=0.\end{eqnarray}$$

We identify $S_{4}$ and $F^{(4)}$ by choosing $X=Z=0$ as the $0$ -section. Let $L_{1},\ldots ,L_{4}$ be four lines defined by

$$\begin{eqnarray}\displaystyle & \displaystyle L_{1}:X+(u^{2}+1)^{2}Y=Z+(v^{2}+1)^{2}W=0, & \displaystyle \nonumber\\ \displaystyle & \displaystyle L_{2}:X+(u^{2}-1)^{2}Y=Z+(v^{2}-1)^{2}W=0, & \displaystyle \nonumber\\ \displaystyle & \displaystyle L_{3}:X+(u^{2}+1)^{2}Y=Z+(v^{2}-1)^{2}W=0, & \displaystyle \nonumber\\ \displaystyle & \displaystyle L_{4}:X+(u^{2}-1)^{2}Y=Z+(v^{2}+1)^{2}W=0. & \displaystyle \nonumber\end{eqnarray}$$

By an abuse of notation, we also denote by $L_{i}$ the corresponding section of $F_{E_{u},E_{v}}^{(4)}$ .

$S_{4}$ may be considered as a family of the intersection of two quadrics. Namely, consider the map $S_{4}\rightarrow \mathbb{P}^{1}$ given by

$$\begin{eqnarray}\unicode[STIX]{x1D708}:(X:Y:Z:W)\mapsto \big(XY+(u^{2}+1)Y^{2}:ZW\big).\end{eqnarray}$$

Over the point $(p:q)\in \mathbb{P}^{1}$ , the fiber of $\unicode[STIX]{x1D708}$ is the intersection of two quadrics

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}rZW=Y(X+(u^{2}+1)^{2}Y),\\ (Z+(v^{2}+1)^{2}W)(Z+(v^{2}-1)^{2}W)=rX(X+(u^{2}-1)^{2}Y),\end{array}\right.\end{eqnarray}$$

where $r=p/q$ . The intersection is a curve of genus 1 for each $r$ , except the following eight values:

$$\begin{eqnarray}r=\frac{\pm 2i}{(u+1)^{2}},\qquad \frac{\pm 2i}{(u-1)^{2}},\qquad \frac{\pm 2iv^{2}}{(u+1)^{2}},\qquad \frac{\pm 2iv^{2}}{(u-1)^{2}}.\end{eqnarray}$$

At each of these values of $r$ , the intersection degenerates and becomes a union of two plane conics.

Let $R_{1},\ldots ,R_{4}$ be one of the plane conics at each of the values

$$\begin{eqnarray}r=2i/(u+1)^{2},\qquad -2i/(u+1)^{2},\qquad 2i/(u-1)^{2},\qquad 2iv^{2}/(u-1)^{2},\end{eqnarray}$$

respectively. Similarly, in the family

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}rZ(Z+(v^{2}-1)^{2}W)=(X+(u^{2}+1)^{2}Y)(X+(u^{2}-1)^{2}Y),\\ W(Z+(v^{2}+1)^{2}W)=rXY,\end{array}\right.\end{eqnarray}$$

let $R_{5},R_{6}$ be one of the conics at the value $r=\pm 2i/(v+1)^{2}$ , and let $R_{7}$ be one of the conics at $r=(u+1)^{2}/(v+1)^{2}$ in the family

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}rZ(Z+(v^{2}-1)^{2}W)=X(X+(u^{2}-1)^{2}Y),\\ W(Z+(v^{2}+1)^{2}W)=rY(X+(u^{2}+1)^{2}Y).\end{array}\right.\end{eqnarray}$$

Explicit choices for $R_{1},\ldots ,R_{7}$ are made in the computer files. Finally, let $R_{8}$ be the section obtained by letting $u\mapsto iu$ in $R_{1}$ .

Proposition 5.5. The basis in Theorem 5.1 and that of Theorem 5.3 are related as follows.

$$\begin{eqnarray}\displaystyle L_{1} & = & \displaystyle -P_{1}+P_{3}-P_{4}+P_{5}-2Q_{1},\nonumber\\ \displaystyle L_{2} & = & \displaystyle -2P_{1}-P_{2}+P_{3}-P_{4}+P_{5}-2Q_{1},\nonumber\\ \displaystyle L_{3} & = & \displaystyle -2P_{1}^{\prime }-P_{2}^{\prime }+P_{3}^{\prime }-P_{4}^{\prime }+P_{5}^{\prime }-2Q_{1}^{\prime },\nonumber\\ \displaystyle L_{4} & = & \displaystyle -P_{1}^{\prime }+P_{3}^{\prime }-P_{4}^{\prime }+P_{5}^{\prime }-2Q_{1}^{\prime },\nonumber\\ \displaystyle R_{1} & = & \displaystyle -P_{1}+P_{3}-Q_{1}-P_{4}^{\prime }-Q_{1}^{\prime },\nonumber\\ \displaystyle R_{2} & = & \displaystyle P_{3}-P_{4}-Q_{1}-P_{1}^{\prime }-P_{4}^{\prime }+P_{5}^{\prime }-Q_{1}^{\prime }\nonumber\\ \displaystyle R_{3} & = & \displaystyle -P_{1}+P_{3}-P_{4}+P_{5}-Q_{1}-P_{1}^{\prime }-P_{2}^{\prime }+P_{5}^{\prime }-Q_{1}^{\prime },\nonumber\\ \displaystyle R_{4} & = & \displaystyle -P_{4}+P_{5}-Q_{1}-P_{1}^{\prime }-P_{2}^{\prime }+P_{3}^{\prime }-Q_{1}^{\prime },\nonumber\\ \displaystyle R_{5} & = & \displaystyle P_{2}-P_{4}-Q_{1}-P_{4}^{\prime }+P_{5}^{\prime }-Q_{1}^{\prime },\nonumber\\ \displaystyle R_{6} & = & \displaystyle -P_{1}+P_{3}-Q_{1}+P_{2}^{\prime }-P_{4}^{\prime }-Q_{1}^{\prime },\nonumber\\ \displaystyle R_{7} & = & \displaystyle -P_{4}-Q_{1}-2P_{1}^{\prime }-P_{2}^{\prime }+P_{3}^{\prime }-P_{4}^{\prime }+2P_{5}^{\prime }-2Q_{1}^{\prime },\nonumber\\ \displaystyle R_{8} & = & \displaystyle -P_{1}-Q_{1}+P_{3}^{\prime }-P_{4}^{\prime }-Q_{1}^{\prime }.\nonumber\end{eqnarray}$$

6.1 Elliptic modular surface associated with $\unicode[STIX]{x1D6E4}(5)$

We begin with the elliptic modular surface over $\unicode[STIX]{x1D6E4}(5)$ , which is described in [Reference Rubin and SilverbergRS1] for instance.

(6.1) $$\begin{eqnarray}\displaystyle y^{2} & = & \displaystyle x^{3}-27(\unicode[STIX]{x1D707}^{20}-228\unicode[STIX]{x1D707}^{15}+494\unicode[STIX]{x1D707}^{10}+228\unicode[STIX]{x1D707}^{5}+1)x\nonumber\\ \displaystyle & & \displaystyle +\,54(\unicode[STIX]{x1D707}^{30}+522\unicode[STIX]{x1D707}^{25}-10005\unicode[STIX]{x1D707}^{20}-10005\unicode[STIX]{x1D707}^{10}-522\unicode[STIX]{x1D707}^{5}+1).\end{eqnarray}$$

This elliptic curve has full $5$ -torsion defined over $\mathbb{Q}(\unicode[STIX]{x1D701})(\unicode[STIX]{x1D707})$ , where $\unicode[STIX]{x1D701}$ is a primitive fifth root of unity. Let $\unicode[STIX]{x1D702}=\unicode[STIX]{x1D701}+\unicode[STIX]{x1D701}^{-1}$ , which can be taken to be the golden ratio $(1+\sqrt{5})/2$ . A basis for the $5$ -torsion is given by

$$\begin{eqnarray}\displaystyle T_{1} & = & \displaystyle \big(3(\unicode[STIX]{x1D707}^{10}+12\unicode[STIX]{x1D707}^{8}-12\unicode[STIX]{x1D707}^{7}+24\unicode[STIX]{x1D707}^{6}+30\unicode[STIX]{x1D707}^{5}+60\unicode[STIX]{x1D707}^{4}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,36\unicode[STIX]{x1D707}^{3}+24\unicode[STIX]{x1D707}^{2}+12\unicode[STIX]{x1D707}+1),\nonumber\\ \displaystyle & & \displaystyle \quad 108\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}^{4}-3\unicode[STIX]{x1D707}^{3}+4\unicode[STIX]{x1D707}^{2}-2\unicode[STIX]{x1D707}+1)(\unicode[STIX]{x1D707}^{4}+2\unicode[STIX]{x1D707}^{3}+4\unicode[STIX]{x1D707}^{2}+3\unicode[STIX]{x1D707}+1)^{2}\big)\nonumber\\ \displaystyle T_{2} & = & \displaystyle \Big(-3((12\unicode[STIX]{x1D702}+19)\unicode[STIX]{x1D707}^{10}+66(2\unicode[STIX]{x1D702}-1)\unicode[STIX]{x1D707}^{5}-12\unicode[STIX]{x1D702}+31)/5,\nonumber\\ \displaystyle & & \displaystyle \quad 108(3\unicode[STIX]{x1D701}^{3}-\unicode[STIX]{x1D701}^{2}+2\unicode[STIX]{x1D701}+1)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\big(\unicode[STIX]{x1D707}^{15}+(-5\unicode[STIX]{x1D702}+19)\unicode[STIX]{x1D707}^{10}+(-55\unicode[STIX]{x1D702}+87)\unicode[STIX]{x1D707}^{5}+5\unicode[STIX]{x1D702}-8\big)/5\Big).\nonumber\end{eqnarray}$$

Let

$$\begin{eqnarray}\unicode[STIX]{x1D703}=\left(\begin{array}{@{}cc@{}}1 & 1\\ 0 & 1\end{array}\right)\qquad \text{and}\qquad \unicode[STIX]{x1D70C}=\left(\begin{array}{@{}cc@{}}0 & 1\\ -1 & 0\end{array}\right)\end{eqnarray}$$

be generators for $G=\operatorname{SL}_{2}(\mathbb{F}_{5})$ , and let $\unicode[STIX]{x1D70B}$ be the representation of $G$ on $\mathbb{P}_{\unicode[STIX]{x1D707}}^{1}$ as follows.

$$\begin{eqnarray}\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}):\unicode[STIX]{x1D707}\mapsto \unicode[STIX]{x1D701}\unicode[STIX]{x1D707},\qquad \unicode[STIX]{x1D70B}(\unicode[STIX]{x1D70C}):\unicode[STIX]{x1D707}\mapsto -1/\unicode[STIX]{x1D707}.\end{eqnarray}$$

The $j$ -invariant of (4.1) is given by

$$\begin{eqnarray}j=-\frac{(\unicode[STIX]{x1D707}^{20}-228\unicode[STIX]{x1D707}^{15}+494\unicode[STIX]{x1D707}^{10}+228\unicode[STIX]{x1D707}^{5}+1)^{3}}{\unicode[STIX]{x1D707}^{5}(\unicode[STIX]{x1D707}^{10}+11\unicode[STIX]{x1D707}^{5}-1)^{5}}\end{eqnarray}$$

and is invariant under the action of the icosahedral group $\unicode[STIX]{x1D70B}(\operatorname{SL}_{2}(\mathbb{F}_{5}))\cong A_{5}$ .

As before, there is a compatible action of $G$ on the universal elliptic curve, given by

$$\begin{eqnarray}\unicode[STIX]{x1D703}:(x,y,\unicode[STIX]{x1D707})\rightarrow (x,y,\unicode[STIX]{x1D701}\unicode[STIX]{x1D707}),\qquad \unicode[STIX]{x1D70C}:(x,y,\unicode[STIX]{x1D707})\rightarrow (x/\unicode[STIX]{x1D707}^{10},y/\unicode[STIX]{x1D707}^{15},-1/\unicode[STIX]{x1D707}),\end{eqnarray}$$

and therefore an action on the Mordell–Weil group ${\mathcal{E}}_{\unicode[STIX]{x1D6E4}(5)}(k(\unicode[STIX]{x1D707}))$ by $(\unicode[STIX]{x1D6FE}P)(\unicode[STIX]{x1D707})=\unicode[STIX]{x1D6FE}\big(P(\unicode[STIX]{x1D6FE}^{-1}\unicode[STIX]{x1D707})\big)$ . On the $5$ -torsion sections, we have

$$\begin{eqnarray}\unicode[STIX]{x1D703}:\left\{\begin{array}{@{}l@{}}T_{1}\mapsto T_{1}+T_{2}\\ T_{2}\mapsto T_{2},\end{array}\right.\qquad \unicode[STIX]{x1D70C}:\left\{\begin{array}{@{}l@{}}T_{1}\mapsto 2T_{1}\\ T_{2}\mapsto -2T_{2}.\end{array}\right.\end{eqnarray}$$

This action is conjugate to the usual linear action of $\operatorname{PSL}_{2}(\mathbb{F}_{5})$ (since $-1$ is a square modulo $5$ , the usual action of $\unicode[STIX]{x1D70C}$ is diagonalizable).

6.2 $F^{(5)}$ for universal families

Now, take two copies of (6.1)

$$\begin{eqnarray}\displaystyle E_{u}:y_{1}^{2} & = & \displaystyle x_{1}^{3}-27(u^{20}-228u^{15}+494u^{10}+228u^{5}+1)x_{1}\nonumber\\ \displaystyle & & \displaystyle +\,54(u^{30}+522u^{25}-10005u^{20}-10005u^{10}-522u^{5}+1),\nonumber\\ \displaystyle E_{v}:y_{2}^{2} & = & \displaystyle x_{2}^{3}-27(v^{20}-228v^{15}+494v^{10}+228v^{5}+1)x_{2}\nonumber\\ \displaystyle & & \displaystyle +\,54(v^{30}+522v^{25}-10005v^{20}-10005v^{10}-522v^{5}+1).\nonumber\end{eqnarray}$$

Computing $\mathit{Ino}(E_{u},E_{v})$ by (2.6) and base changing, we get the following Weierstrass equation for $F_{E_{u},E_{v}}^{(5)}$ :

(6.2) $$\begin{eqnarray}\displaystyle F_{E_{u},E_{v}}^{(5)}:Y^{2} & = & \displaystyle X^{3}-27(u^{20}-228u^{15}+494u^{10}+228u^{5}+1)\nonumber\\ \displaystyle & & \displaystyle \qquad \qquad \times \,(v^{20}-228v^{15}+494v^{10}+228v^{5}+1)X\nonumber\\ \displaystyle & & \displaystyle -\,2^{6}3^{6}u^{5}(u^{10}+11u^{5}-1)^{5}t_{5}^{5}-\frac{2^{6}3^{6}v^{5}(v^{10}+11v^{5}-1)^{5}}{t_{5}^{5}}\nonumber\\ \displaystyle & & \displaystyle +\,54(u^{30}+522u^{25}-10005u^{20}-10005u^{10}-522u^{5}+1)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,(v^{30}+522v^{25}-10005v^{20}-10005v^{10}-522v^{5}+1).\end{eqnarray}$$

(Here we have scaled $X$ and $Y$ by $9$ and $27$ respectively, to make the coefficients smaller.)

As before, let ${\mathcal{E}}_{u}\rightarrow \mathbb{P}_{u}^{1}$ and ${\mathcal{E}}_{v}\rightarrow \mathbb{P}_{v}^{1}$ elliptic modular surfaces associated with $E_{u}$ and $E_{v}$ , respectively. Also let $G_{u}=\langle \unicode[STIX]{x1D703}_{u},\unicode[STIX]{x1D70C}_{u}\rangle$ and $G_{v}=\langle \unicode[STIX]{x1D703}_{v},\unicode[STIX]{x1D70C}_{v}\rangle$ be groups of automorphisms of ${\mathcal{E}}_{u}$ and ${\mathcal{E}}_{v}$ described above, respectively. We consider (6.2) as the family of elliptic surfaces ${\mathcal{F}}_{E_{u},E_{v}}^{(5)}\rightarrow \mathbb{P}_{u}^{1}\times \mathbb{P}_{v}^{1}$ parametrized by $u$ and $v$ . The total space is a fourfold.

Analogously to 4.3, we have an action of $G_{u}\times G_{v}$ on this variety, with the corresponding expressions being somewhat simpler:

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D703}_{u}:(X,Y,t_{5},u,v) & \rightarrow & \displaystyle (X,Y,t_{5},\unicode[STIX]{x1D701}u,v)\nonumber\\ \displaystyle \unicode[STIX]{x1D703}_{v}:(X,Y,t_{5},u,v) & \rightarrow & \displaystyle (X,Y,t_{5},u,\unicode[STIX]{x1D701}v)\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{u}:(X,Y,t_{5},u,v) & \rightarrow & \displaystyle \left(\frac{X}{u^{10}},\frac{Y}{u^{15}},t_{5}u^{6},-\frac{1}{u},v\right)\nonumber\\ \displaystyle \unicode[STIX]{x1D70C}_{v}:(X,Y,t_{5},u,v) & \rightarrow & \displaystyle \left(\frac{X}{v^{10}},\frac{Y}{v^{15}},\frac{t_{5}}{v^{6}},u,-\frac{1}{v}\right).\nonumber\end{eqnarray}$$

There is also the action of the dihedral group $D_{10}$ on $F^{(5)}$ . Let $D_{10}=\langle \unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\rangle$ , with $\unicode[STIX]{x1D70E}^{5}=1,\unicode[STIX]{x1D70F}^{2}=1$ and $\unicode[STIX]{x1D70E}\unicode[STIX]{x1D70F}=\unicode[STIX]{x1D70F}\unicode[STIX]{x1D70E}^{-1}$ . Then $\unicode[STIX]{x1D70E}$ acts by twisting $t_{5}$ by $\unicode[STIX]{x1D701}$ and $\unicode[STIX]{x1D70F}$ by taking it to $\unicode[STIX]{x1D6FF}/t$ , where

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}=v(v^{10}+11v^{5}-1)/\big(u(u^{10}+11u^{5}-1)\big).\end{eqnarray}$$

Consequently, we have the following result, which describes the action of these groups on the sections.

Proposition 6.1. The group $D_{10}\times G_{u}\times G_{v}$ acts on the group of the sections $F_{E_{u},E_{v}}^{(5)}(\bar{k}(u,v)(t_{5}))$ as follows:

$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D70E}\big(X(t_{5},u,v),Y(t_{5},u,v)\big)=\big(X(\unicode[STIX]{x1D701}^{-1}t_{5},u,v),Y(\unicode[STIX]{x1D701}^{-1}t_{5},u,v)\big) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D70F}\big(X(t_{5},u,v),Y(t_{5},u,v)\big)=\big(X(\unicode[STIX]{x1D6FF}/t_{5},u,v),Y(\unicode[STIX]{x1D6FF}/t_{5},u,v)\big) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D703}_{u}\big(X(t_{5},u,v),Y(t_{5},u,v)\big)=\big(X(t_{5},\unicode[STIX]{x1D701}^{-1}u,v),Y(t_{5},\unicode[STIX]{x1D701}^{-1}u,v)\big) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D703}_{v}\big(X(t_{5},u,v),Y(t_{5},u,v)\big)=\big(X(t_{5},u,\unicode[STIX]{x1D701}^{-1}v),Y(t_{5},u,\unicode[STIX]{x1D701}^{-1}v)\big) & \displaystyle \nonumber\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D70C}_{u}\big(X(t_{5},u,v),Y(t_{5},u,v)\big)\nonumber\\ \displaystyle & & \displaystyle \qquad =\big(u^{10}X(t_{5}u^{6},-1/u,v),-u^{15}Y(t_{5}u^{6},-1/u,v)\big)\nonumber\\ \displaystyle & & \displaystyle \unicode[STIX]{x1D70C}_{v}\big(X(t_{5},u,v),Y(t_{5},u,v)\big)\nonumber\\ \displaystyle & & \displaystyle \qquad =\big(v^{10}X(t_{5}/v^{6},u,-1/v),-v^{15}Y(t_{5}/v^{6},u,-1/v)\big).\nonumber\end{eqnarray}$$

6.3 The rational elliptic surface

Set

$$\begin{eqnarray}s=u(u^{10}+11u^{5}-1)t_{5}+\frac{v(v^{10}+11v^{5}-1)}{t_{5}}.\end{eqnarray}$$

Then the equation for $F^{(5)}$ transforms to

$$\begin{eqnarray}R_{s}^{(5)}:Y^{2}=X^{3}-27AX-2^{6}3^{6}(s^{5}-5Bs^{3}+5B^{2}s)+C,\end{eqnarray}$$

where

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}A=(u^{20}-228u^{15}+494u^{10}+228u^{5}+1)\\ \qquad \times \,(v^{20}-228v^{15}+494v^{10}+228v^{5}+1),\\ B=uv(u^{10}+11u^{5}-1)(v^{10}+11v^{5}-1),\\ C=54(u^{30}+522u^{25}-10005u^{20}-10005u^{10}-522u^{5}+1)\\ \qquad \times \,(v^{30}+522v^{25}-10005v^{20}-10005v^{10}-522v^{5}+1).\end{array}\right.\end{eqnarray}$$

This equation defines a rational elliptic surface $R_{s}^{(5)}$ over $K=k(u,v)$ , fibered over $\mathbb{P}_{s}^{1}$ , with the property that its base change to $\mathbb{P}_{t}^{1}$ is $F^{(5)}$ . It has a singular fiber of additive reduction (generically type $\text{II}$ ) at $s=\infty$ , so using Shioda’s specialization technique, we can readily determine an equation of degree $240$ whose roots give all the specializations of the minimal vectors of the Mordell–Weil lattice, which is generically $E_{8}$ . As before, we obtain an action of $\operatorname{PSL}_{2}(\mathbb{F}_{5})$ on the sections by the following lemma, whose (easy) proof is omitted.

Lemma 6.2. The automorphisms $\unicode[STIX]{x1D703}_{u}\unicode[STIX]{x1D703}_{v}$ and $\unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x1D70C}_{v}$ induce automorphisms of the Mordell–Weil group $R_{s}^{(5)}(K(s))$ , given by

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D703}_{u}\unicode[STIX]{x1D703}_{v}:\big(X(s,u,v),Y(s,u,v)\big)\nonumber\\ \displaystyle & & \displaystyle \quad \mapsto \big(X(\unicode[STIX]{x1D701}^{-1}s,\unicode[STIX]{x1D701}^{-1}u,\unicode[STIX]{x1D701}^{-1}v),Y(\unicode[STIX]{x1D701}^{-1}s,\unicode[STIX]{x1D701}^{-1}u,\unicode[STIX]{x1D701}^{-1}v)\big),\nonumber\\ \displaystyle & & \displaystyle \unicode[STIX]{x1D70C}_{u}\unicode[STIX]{x1D70C}_{v}:\big(X(s,u,v),Y(s,u,v)\big)\nonumber\\ \displaystyle & & \displaystyle \quad \mapsto \left((uv)^{10}X\Big(\frac{s}{(uv)^{6}},-\frac{1}{u},-\frac{1}{v}\Big),(uv)^{15}Y\Big(\frac{s}{(uv)^{6}},-\frac{1}{u},-\frac{1}{v}\Big)\right).\nonumber\end{eqnarray}$$

For the rational elliptic surface $R_{s}^{(5)}$ , it follows from general structural results (see [Reference ShiodaSh2]) that the Mordell–Weil lattice is $E_{8}$ , and is spanned by the $240$ smallest vectors. These correspond to sections of the form

$$\begin{eqnarray}\big(X(s),Y(s)\big)=(x_{2}s^{2}+x_{1}s+x_{0},y_{3}s^{3}+y_{2}s^{2}+y_{1}s+y_{0}).\end{eqnarray}$$

The specialization of such a section at $s=\infty$ is $z=x_{2}/y_{3}\in \mathbb{G}_{a}$ . Substituting the above expression for $X$ and $Y$ into the Weierstrass equation, we obtain a system of equations, in which we can eliminate all variables but $z$ . The resulting equation has degree $240$ , and it splits over the field $k(u,v)$ into linear factors. The variables $x_{2},\ldots ,y_{0}$ are rational functions of $z$ with coefficients polynomials in the Weierstrass coefficients. As a result, every section is defined over the field $k(u,v)$ . In the result below, we will just write down the specializations of the relevant sections (the entire expression can be found in the auxiliary source files).

Proposition 6.3. Let $P_{0}$ and $Q_{0}$ be sections whose specializations are given by

$$\begin{eqnarray}\displaystyle z(P_{0}) & = & \displaystyle (\unicode[STIX]{x1D701}+1)\big(uv-(\unicode[STIX]{x1D701}^{2}+1)u-\unicode[STIX]{x1D701}^{2}(\unicode[STIX]{x1D701}^{2}+1)v-\unicode[STIX]{x1D701}^{4}\big)/6,\nonumber\\ \displaystyle z(Q_{0}) & = & \displaystyle \unicode[STIX]{x1D701}^{3}(\unicode[STIX]{x1D701}-1)(u-\unicode[STIX]{x1D701}^{3}v)/6.\nonumber\end{eqnarray}$$

Then $P_{0},\unicode[STIX]{x1D703}(P_{0}),\unicode[STIX]{x1D703}^{-1}(P_{0}),\unicode[STIX]{x1D70C}(P_{0}),\unicode[STIX]{x1D703}^{3}(P_{0}),\unicode[STIX]{x1D703}\unicode[STIX]{x1D70C}(P_{0}),Q_{0},\unicode[STIX]{x1D70C}(Q_{0})$ form a basis of the Mordell–Weil group of $R_{s}^{(5)}$ .

Proof. By direct calculation, the intersection matrix of these sections is

$$\begin{eqnarray}\left(\begin{array}{@{}crcrcccc@{}}2 & 1 & 1 & 0 & 0 & 1 & 1 & 0\\ 1 & 2 & 0 & -1 & 0 & 0 & 1 & 0\\ 1 & 0 & 2 & 1 & 1 & 1 & 1 & 0\\ 0 & -1 & 1 & 2 & 1 & 1 & 0 & 1\\ 0 & 0 & 1 & 1 & 2 & 0 & 1 & 0\\ 1 & 0 & 1 & 1 & 0 & 2 & 0 & 1\\ 1 & 1 & 1 & 0 & 1 & 0 & 2 & 0\\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 2\end{array}\right).\end{eqnarray}$$

Therefore, they span an even unimodular eight-dimensional lattice, which must be $E_{8}$ . Hence, these sections are a basis of the entire Mordell–Weil group.◻

6.4 $F^{(5)}(\overline{k}(t_{5}))$ in the generic case

We now describe the Mordell–Weil group of $F^{(5)}$ in the generic case, when $E_{u}$ and $E_{v}$ are not isogenous.

7.1 Elliptic modular surface associated with $\unicode[STIX]{x1D6E4}(6)$

The modular curve $X(6)$ associated with the congruence subgroup $\unicode[STIX]{x1D6E4}(6)$ is known to be a curve of genus 1 with affine model given by $r^{2}=f^{3}+1$ , and the elliptic modular surface associated with it is given by

(7.1) $$\begin{eqnarray}y^{2}=x^{3}-2(f^{6}+20f^{3}-8)x^{2}+f^{3}(f^{3}-8)^{3}x\end{eqnarray}$$

(cf. [Reference Rubin and SilverbergRS2]). The subgroup of $3$ -torsion points are generated by

$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle (f^{2}(f^{2}+2f+4)^{2},4f^{2}(f^{2}-f+1)(f^{2}+2f+4)^{2})\qquad \text{and}\qquad \nonumber\\ \displaystyle & & \displaystyle \displaystyle (-(f^{3}-8)^{2}/3,4(2\unicode[STIX]{x1D714}+1)(f^{3}+1)(f^{3}-8)^{2}/9),\nonumber\end{eqnarray}$$

and the points of order  $2$ are given by

$$\begin{eqnarray}(0,0),\qquad ((r+1)(r-3)^{3},0),\qquad ((r+1)(r-3)^{3},0).\end{eqnarray}$$

The $j$ -invariant of (7.1) is given by

$$\begin{eqnarray}j=\frac{(f^{3}+4)^{3}(f^{9}+228f^{6}+48f^{3}+64)^{3}}{f^{6}(f^{3}+1)^{3}(f^{3}-8)^{6}}.\end{eqnarray}$$

Note that the curve (7.1) can be written in terms of $r$ :

(7.2) $$\begin{eqnarray}y^{2}=x(x-(r+1)(r-3)^{3})(x-(r-1)(r+3)^{3}),\end{eqnarray}$$

and after scaling $x$ and $y$ it is transformed to

$$\begin{eqnarray}(r-1)(r+3)y^{2}=x(x-1)(x-\unicode[STIX]{x1D706}),\quad \text{where}~\unicode[STIX]{x1D706}=\frac{(r+1)(r-3)^{3}}{(r-1)(r+3)^{3}}.\end{eqnarray}$$

If we view (7.1) as an elliptic surface over $\mathbb{P}_{f}^{1}$ , it is an elliptic modular surface corresponding to $\unicode[STIX]{x1D6E4}(3)\cap \unicode[STIX]{x1D6E4}_{1}(2)$ , whereas (7.2) as an elliptic surface over $\mathbb{P}_{r}^{1}$ is an elliptic modular surface corresponding to $\unicode[STIX]{x1D6E4}(2)\cap \unicode[STIX]{x1D6E4}_{1}(3)$ . Note that the map to $X(2)$ is just $(r,f)\rightarrow \unicode[STIX]{x1D706}$ , whereas the map to $X(3)$ is $(r,f)\rightarrow \unicode[STIX]{x1D707}=(f^{3}+4)/3f^{2}$ .

7.2 $F^{(6)}$ for universal family

We take two copies of the modular curve $X(6)$

$$\begin{eqnarray}r^{2}=f^{3}+1,\qquad q^{2}=g^{3}+1,\end{eqnarray}$$

and the elliptic modular surface (7.1):

$$\begin{eqnarray}\displaystyle E_{r,f}:y_{1}^{2} & = & \displaystyle x_{1}^{3}-2(f^{6}+20f^{3}-8)x_{1}^{2}+f^{3}(f^{3}-8)^{3}x_{1}.\nonumber\\ \displaystyle E_{q,g}:y_{2}^{2} & = & \displaystyle x_{2}^{3}-2(g^{6}+20g^{3}-8)x_{2}^{2}+g^{3}(g^{3}-8)^{3}x_{2}.\nonumber\end{eqnarray}$$

We then obtain $F_{E_{r,f},E_{q,g}}^{(6)}$

(7.3) $$\begin{eqnarray}\displaystyle Y^{2} & = & \displaystyle X^{3}-27(f^{12}+232f^{9}+960f^{6}+256f^{3}+256)\nonumber\\ \displaystyle & & \displaystyle \qquad \times (g^{12}+232g^{9}+960g^{6}+256g^{3}+256)X\nonumber\\ \displaystyle & & \displaystyle +\,54\Bigl(864f^{6}(f^{3}+1)^{3}(f^{3}-8)^{6}t^{6}+\frac{864g^{6}(g^{3}+1)^{3}(g^{3}-8)^{6}}{t^{6}}\nonumber\\ \displaystyle & & \displaystyle +\,(f^{18}-516f^{15}-12072f^{12}-24640f^{9}-30720f^{6}+6144f^{3}+4096)\nonumber\\ \displaystyle & & \displaystyle \times \,(g^{18}-516g^{15}-12072g^{12}-24640g^{9}-30720g^{6}+6144g^{3}+4096)\Bigr).\end{eqnarray}$$

7.3 Rational elliptic surfaces with parameter $s_{6,i}$

As before, we have several rational elliptic surfaces arising as quotients of $F^{(6)}$ . Namely, for $0\leqslant i\leqslant 6$ , let

$$\begin{eqnarray}s_{6,i}=s_{i}=\unicode[STIX]{x1D701}_{6}rf(f^{3}-8)t+\frac{qg(g^{3}-8)}{t},\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{6}=-\unicode[STIX]{x1D714}$ is a primitive sixth root of unity. Then the equation transforms to the rational elliptic surface

$$\begin{eqnarray}R_{s}^{(6)}:Y^{2}=X^{3}-AX+6^{6}(s^{6}-6Bs^{4}+9B^{2}s^{2}-2B^{3})+C\end{eqnarray}$$

where $s=s_{0}=s_{6,0}$ and

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}A=27(f^{12}+232f^{9}+960f^{6}+256f^{3}+256)\\ \qquad \times \,(g^{12}+232g^{9}+960g^{6}+256g^{3}+256)\\ B=qrfg(f^{3}-8)(g^{3}-8)\\ C=54(f^{18}-516f^{15}-12072f^{12}-24640f^{9}-30720f^{6}+6144f^{3}+4096)\\ \qquad \times \,(g^{18}-516g^{15}-12072g^{12}-24640g^{9}-30720g^{6}+6144g^{3}+4096).\end{array}\right.\end{eqnarray}$$

The Mordell–Weil lattice of this elliptic surface is generically $E_{8}$ . It has generically twelve $\text{I}_{1}$ fibers, none of them defined over the ground field $k(u,v)$ . Therefore, one has to proceed by brute force in order to compute a basis of sections. Taking $X$ to be a quadratic polynomial in $s$ , and $Y$ cubic, with undetermined coefficients, we obtain a system of equations for the coefficients. The $240$ solutions give the sections of minimal height. Since these are complicated to write down, we will not do so here. The formulas may be found in the auxiliary files. Instead, we will give a more conceptual description below, in terms of sections arising from $F^{(3)}$ and its twist, the cubic surface.

We may also form the rational elliptic surface in terms of $s_{1}$ : the equation becomes (with the same values of $A,B,C$ as above):

$$\begin{eqnarray}R_{s_{1}}^{(6)}:Y^{2}=X^{3}-AX+6^{6}(s_{1}^{6}-6\unicode[STIX]{x1D701}Bs_{1}^{4}+9\unicode[STIX]{x1D701}^{2}B^{2}s_{1}^{2}-2\unicode[STIX]{x1D701}^{3}B^{3})+C.\end{eqnarray}$$

A similar calculation gives the $240$ minimal height sections for this elliptic surface. Let $P_{1},\ldots ,P_{8}$ be the sections coming from $E_{s}$ and $P_{1}^{\prime },\ldots ,P_{8}^{\prime }$ those from $E_{s_{1}}$ .

7.4 $F^{(6)}$ as a double cover of a cubic surface

Now we describe a different method to compute the Mordell–Weil group of $F^{(6)}$ , going through its quotient $F^{(3)}$ and a quadratic twist of this quotient, which is a rational surface. In the remainder of this subsection, we will let $\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D707}$ be parameters on $X(2)$ . For an elliptic curve $E$ over $k(t)$ , we denote by $^{t}\!E$ its quadratic twist.

Lemma 7.3. Let $E_{1}$ and $E_{2}$ be given as in (2.1). The Kodaira–Néron model of the quadratic twist $^{t_{3}}\!F_{E_{1},E_{2}}^{(3)}$ is birationally equivalent to the cubic surface given by

$$\begin{eqnarray}Z^{3}+cZW^{2}+dW^{3}=X^{3}+aXY^{2}+bY^{3}.\end{eqnarray}$$

Proof. The equation of $^{t_{3}}\!F^{(3)}$ is given by

(7.4) $$\begin{eqnarray}t_{3}Y^{2}=X^{3}-3\,ac\,X+\frac{1}{64}\Bigl(\unicode[STIX]{x1D6E5}_{E_{1}}t_{3}^{3}+864\,bd+\frac{\unicode[STIX]{x1D6E5}_{E_{2}}}{t_{3}^{3}}\Bigr).\end{eqnarray}$$

Since $t_{3}=t_{6}^{2}$ , this equation can be written as

(7.5) $$\begin{eqnarray}(t_{6}Y)^{2}=X^{3}-3\,ac\,X+\frac{1}{64}\Bigl(\unicode[STIX]{x1D6E5}_{E_{1}}t_{6}^{6}+864\,bd+\frac{\unicode[STIX]{x1D6E5}_{E_{2}}}{t_{6}^{6}}\Bigr).\end{eqnarray}$$

Rewriting the change of coordinates (2.4) using $t_{6}^{2}=t_{3}$ , we see that $X$ and $t_{6}Y$ are written in terms of $t_{3}$ :

$$\begin{eqnarray}\left\{\begin{array}{@{}l@{}}\displaystyle \hspace{-1.0pt}X=\frac{-t_{3}(2at_{3}^{2}-c)x_{1}-3(bt_{3}^{3}-d)-(at_{3}^{2}-2c)x_{2}}{t_{3}(t_{3}x_{1}-x_{2})},\\ \displaystyle \hspace{-1.0pt}t_{6}Y=\frac{6(at_{3}^{2}\hspace{-1.0pt}-\hspace{-1.0pt}c)(bt_{3}^{3}\hspace{-1.0pt}-\hspace{-1.0pt}d)+6(at_{3}^{2}\hspace{-1.0pt}-\hspace{-1.0pt}c)(at_{3}^{3}x_{1}\hspace{-1.0pt}-\hspace{-1.0pt}cx_{2})-9(bt_{3}^{3}\hspace{-1.0pt}-\hspace{-1.0pt}d)(t_{3}^{2}x_{1}^{2}\hspace{-1.0pt}-\hspace{-1.0pt}x_{2}^{2})}{2t_{3}(t_{3}x_{1}\hspace{-1.0pt}-\hspace{-1.0pt}x_{2})^{2}}.\end{array}\right.\end{eqnarray}$$

Plugging these back into (7.5), we obtain the equation

$$\begin{eqnarray}x_{2}^{3}+cx_{2}+d=t_{3}^{3}(x_{1}^{3}+ax_{1}+b).\end{eqnarray}$$

Now, if we let $x_{1}=X/Y$ , $x_{2}=Z/W$ and $t_{3}=Y/W$ , we obtain the desired homogeneous cubic equation.◻

We will use the following well-known lemma to put together the sections from the quotients $F^{(3)}$ and $^{t_{3}}\!F^{(3)}$ .

Proof. Let $\unicode[STIX]{x1D704}$ be the automorphism $\sqrt{T}\mapsto -\sqrt{T}$ . Then the composition of the maps

$$\begin{eqnarray}\displaystyle E(k(\sqrt{T}))\longrightarrow E(k(T))\oplus ^{T}\!E(k(T)) & \longrightarrow & \displaystyle E(k(\sqrt{T}))\nonumber\\ \displaystyle P\longmapsto (P+\unicode[STIX]{x1D704}(P),P-\unicode[STIX]{x1D704}(P)) & & \displaystyle \nonumber\\ \displaystyle (Q,R) & \longmapsto & \displaystyle Q+R\nonumber\end{eqnarray}$$

is the multiplication-by- $2$ map $[2]$ . Since the image of $[2]$ is a subgroup of finite index, the assertion follows.◻

In order to compute $^{t_{3}}\!F^{(3)}(\bar{k}(t_{3}))$ , we find the twenty-seven lines contained in the cubic surface. To state the results clearly, we take $E_{1}$ and $E_{2}$ to be in Legendre form as in (3.4), and consider the cubic surface

(7.6) $$\begin{eqnarray}Z(Z-W)(Z-\unicode[STIX]{x1D707}W)=X(X-Y)(X-\unicode[STIX]{x1D706}Y).\end{eqnarray}$$

As is well known, the group generated by $\unicode[STIX]{x1D706}\mapsto 1-\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D706}\mapsto 1/\unicode[STIX]{x1D706}$ leave the $j$ -invariant of $y^{2}=x(x-1)(x-\unicode[STIX]{x1D706})$ . As in Proposition 4.3, this action lifts to the family of cubic surfaces.

Proposition 7.5. There are automorphisms acting on the family of cubic surfaces (7.6) parametrized by $(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$ :

$$\begin{eqnarray}\displaystyle & \displaystyle ((X:Y:Z:W),\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})\mapsto ((X-Y:-Y:Z:W),1-\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle ((X:Y:Z:W),\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})\mapsto ((X:\unicode[STIX]{x1D706}Y:Z:W),1/\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle ((X:Y:Z:W),\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})\mapsto ((X:Y:Z-W:-W),\unicode[STIX]{x1D706},1-\unicode[STIX]{x1D707}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle ((X:Y:Z:W),\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})\mapsto ((X:Y:Z:\unicode[STIX]{x1D707}W),\unicode[STIX]{x1D706},1/\unicode[STIX]{x1D707}). & \displaystyle \nonumber\end{eqnarray}$$

These automorphism acts on the set of twenty-seven lines contained in (7.6).

Let $\unicode[STIX]{x1D6FF}=(\unicode[STIX]{x1D6E5}_{E_{2}}/\unicode[STIX]{x1D6E5}_{E_{1}})^{1/6}=(\unicode[STIX]{x1D707}(\unicode[STIX]{x1D707}-1))^{1/3}/(\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1))^{1/3}$ . Note $\unicode[STIX]{x1D6FF}^{3}$ is invariant under $\unicode[STIX]{x1D706}\rightarrow 1-\unicode[STIX]{x1D706}$ and $\unicode[STIX]{x1D707}\rightarrow 1-\unicode[STIX]{x1D707}$ , whereas under $\unicode[STIX]{x1D706}\rightarrow 1/\unicode[STIX]{x1D706}$ , it is taken to $-\unicode[STIX]{x1D6FF}^{3}/\unicode[STIX]{x1D706}^{3}=(-\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D706})^{3}$ . Therefore, we may extend the action of the group of automorphisms to $\unicode[STIX]{x1D6FF}$ in a natural way. The next result shows that all the twenty-seven lines are defined over the cubic extension field of $k(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})$ defined by $\unicode[STIX]{x1D6FF}$ .

Proposition 7.6. The twenty-seven lines contained in the cubic surface (7.6) are given as follows. In terms of homogeneous parameters $(\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FD})$ on $\mathbb{P}^{1}$ , the lines $(X,Y,Z,W)$ belong to the following list:

$$\begin{eqnarray}\displaystyle & \displaystyle (0:\unicode[STIX]{x1D6FC}:0:\unicode[STIX]{x1D6FD}),\qquad (\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FC}:0:\unicode[STIX]{x1D6FD}),\qquad (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FC}:0:\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (0:\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6FD}),\qquad (\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6FD}),\qquad (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (0:\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6FD}),\qquad (\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6FD}),\qquad (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D707}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D707})\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D707})\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D707}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}+\unicode[STIX]{x1D706}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D707})\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D707})\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D707}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706})\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D707})\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D707})\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}(\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}:-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}+1)\unicode[STIX]{x1D6FC}:-\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}+1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}:-\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}+1)\unicode[STIX]{x1D6FC}:-\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}+1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FC}:-\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}-1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}+1)\unicode[STIX]{x1D6FC}:-\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D706}+1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}+1)\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}+1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}+1)\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}+1)\unicode[STIX]{x1D6FD}), & \displaystyle \nonumber\\ \displaystyle & \displaystyle (\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FC}-\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FF}^{2}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}+1)\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D706}-1)\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FD}:(\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}-\unicode[STIX]{x1D707}+1)\unicode[STIX]{x1D6FD}). & \displaystyle \nonumber\end{eqnarray}$$

They may be generated by taking the orbits of the first and the last three lines in the list, under the action of the automorphism group defined by Proposition 7.5.

Proof. The first nine lines are obvious ones; they are obtained by letting one of the factors of the left hand side of the equation (7.6) equal $0$ and one of the right hand side equal $0$ . The other eighteen lines are obtained as follows. Take a factor from the left hand side and another from the right hand side, say $Z-Y$ and $X-\unicode[STIX]{x1D706}W$ . Take a parameter $m$ and let $Z-Y=m(X-\unicode[STIX]{x1D706}W)$ . We will take the intersections of the cubic surface with this family of planes. By construction, they always contain the line $Z-Y=X-\unicode[STIX]{x1D706}W=0$ . The family of residual conics will degenerate to pairs of lines at suitable values of $m$ .

Concretely, we replace $Z$ by $m(X-Y)+W$ in the equation of the surface, and we obtain a family of conics in $X,Y,W$ :

$$\begin{eqnarray}\displaystyle & & \displaystyle (m^{3}-1)X^{2}-(\unicode[STIX]{x1D707}-1)mW^{2}+m^{3}Y^{2}-(\unicode[STIX]{x1D707}-2)m^{2}XW\nonumber\\ \displaystyle & & \displaystyle \qquad -(2m^{3}-\unicode[STIX]{x1D706})XY+(\unicode[STIX]{x1D707}-2)m^{2}YW=0.\nonumber\end{eqnarray}$$

Writing this equation in matrix form:

$$\begin{eqnarray}\left(\begin{array}{@{}ccc@{}}X & Y & W\end{array}\right)\left(\begin{array}{@{}ccc@{}}m^{3}-1 & -m^{3}+\unicode[STIX]{x1D706}/2 & -(\unicode[STIX]{x1D707}-1)m^{2}/2\\ -m^{3}+\unicode[STIX]{x1D706}/2 & m^{3} & (\unicode[STIX]{x1D707}-2)m^{2}/2\\ -(\unicode[STIX]{x1D707}-1)m^{2}/2 & (\unicode[STIX]{x1D707}-2)m^{2}/2 & -(\unicode[STIX]{x1D707}-1)m\end{array}\right)\left(\begin{array}{@{}c@{}}X\\ Y\\ W\end{array}\right)=0.\end{eqnarray}$$

We then calculate the determinant of this matrix:

$$\begin{eqnarray}-(\unicode[STIX]{x1D706}-1)m(\unicode[STIX]{x1D707}m-\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF})(\unicode[STIX]{x1D707}m-\unicode[STIX]{x1D714}\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF})(\unicode[STIX]{x1D707}m-\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF})/(4\unicode[STIX]{x1D707}).\end{eqnarray}$$

So, at $m=0,\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D707},\unicode[STIX]{x1D714}\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D707},\unicode[STIX]{x1D714}^{2}\unicode[STIX]{x1D706}\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D707}$ , the conic becomes a pair of lines. We repeat this process, and eliminate the duplicates to obtain the list of all the twenty-seven lines.◻

The elliptic surface $^{t_{3}}F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(3)}$ is obtained from the family of plane cubic curves

$$\begin{eqnarray}x_{2}(x_{2}-z)(x_{2}-\unicode[STIX]{x1D707}z)=t_{3}^{3}x_{1}(x_{1}-z)(x_{1}-\unicode[STIX]{x1D706}z)\end{eqnarray}$$

and the rational point $(x_{1}:x_{2}:z)=(1:t_{3}:0)$ . Let us recall its equation (a specialization of (7.4), where the elliptic curves are given by (7.6)):

$$\begin{eqnarray}\displaystyle ty^{2} & = & \displaystyle x^{3}-27(\unicode[STIX]{x1D706}^{2}-\unicode[STIX]{x1D706}+1)(\unicode[STIX]{x1D707}^{2}-\unicode[STIX]{x1D707}+1)x\nonumber\\ \displaystyle & & \displaystyle +\,729\unicode[STIX]{x1D706}^{2}(\unicode[STIX]{x1D706}-1)^{2}t^{3}/4+729\unicode[STIX]{x1D707}^{2}(\unicode[STIX]{x1D707}-1)^{2}/(4t^{3})\nonumber\\ \displaystyle & & \displaystyle +\,27(\unicode[STIX]{x1D706}-2)(\unicode[STIX]{x1D706}+1)(2\unicode[STIX]{x1D706}-1)(\unicode[STIX]{x1D707}-2)(\unicode[STIX]{x1D707}+1)(2\unicode[STIX]{x1D707}-1)/2.\nonumber\end{eqnarray}$$

There are two other rational points $(1:\unicode[STIX]{x1D714}t_{3}:0)$ and $(1:\unicode[STIX]{x1D714}^{2}t_{3}:0)$ , which become sections of $^{t_{3}}F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(3)}$ given by

$$\begin{eqnarray}\displaystyle R_{1} & = & \displaystyle \Bigl(-3(\unicode[STIX]{x1D714}^{2}(\unicode[STIX]{x1D706}^{2}-\unicode[STIX]{x1D706}+1)t^{2}+\unicode[STIX]{x1D714}(\unicode[STIX]{x1D707}^{2}-\unicode[STIX]{x1D707}+1)),\nonumber\\ \displaystyle & & \displaystyle \quad 3(2\unicode[STIX]{x1D714}+1)((\unicode[STIX]{x1D706}+1)(\unicode[STIX]{x1D706}-2)(2\unicode[STIX]{x1D706}-1)t^{3}-(\unicode[STIX]{x1D707}+1)(\unicode[STIX]{x1D707}-2)(2\unicode[STIX]{x1D707}-1))/2\Bigr),\nonumber\\ \displaystyle R_{2} & = & \displaystyle \Bigl(-3(\unicode[STIX]{x1D714}(\unicode[STIX]{x1D706}^{2}-\unicode[STIX]{x1D706}+1)t^{2}+\unicode[STIX]{x1D714}^{2}(\unicode[STIX]{x1D707}^{2}-\unicode[STIX]{x1D707}+1)),\nonumber\\ \displaystyle & & \displaystyle \quad 3(2\unicode[STIX]{x1D714}+1)((\unicode[STIX]{x1D706}+1)(\unicode[STIX]{x1D706}-2)(2\unicode[STIX]{x1D706}-1)t^{3}-(\unicode[STIX]{x1D707}+1)(\unicode[STIX]{x1D707}-2)(2\unicode[STIX]{x1D707}-1))/2\Bigr).\nonumber\end{eqnarray}$$

The elliptic surface $^{t_{3}}F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(3)}$ can be obtained by blowing up the cubic surface (7.6) at the three points which correspond to these, namely,

$$\begin{eqnarray}(X:Y:Z:W)=(1:0:1:0),(1:0:\unicode[STIX]{x1D714}:0),\text{ and }(1:0:\unicode[STIX]{x1D714}^{2}:0).\end{eqnarray}$$

Remark 7.8. The sections $R_{1}$ and $R_{2}$ above, along with the sections coming from lines $1$ , $2$ , $4$ , $5$ , $10$ and $12$ , form a basis of the Mordell–Weil lattice. Below, we display some of these sections; the remaining ones are omitted for lack of space. The formulas for the full basis may be obtained from the auxiliary files.

$$\begin{eqnarray}\displaystyle R_{3} & := & \displaystyle L_{1}=\Big(3\big(3lt^{2}+(l+1)(m+1)t+3m\big)/t,\nonumber\\ \displaystyle & & \displaystyle \quad \qquad -\,27\big(l(l+1)t^{3}+2l(m+1)t^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \qquad +\,2(l+1)mt+m(m+1)\big)/(2t^{2})\Big)\nonumber\\ \displaystyle R_{4} & := & \displaystyle L_{2}=\Big(-3\big(3(l-1)t^{2}-(l-2)(m+1)t-3m\big)/t,\nonumber\\ \displaystyle & & \displaystyle \quad \qquad 27\big((l-1)(l-2)t^{3}+2(l-1)(m+1)t^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \qquad -\,2(l-2)mt-m(m+1)\big)/(2t^{2})\Big)\nonumber\\ \displaystyle R_{5} & := & \displaystyle L_{4}=\Big(3\big(3lt^{2}+(l+1)(m-2)t-3(m-1)\big)/t,\nonumber\\ \displaystyle & & \displaystyle \quad \qquad -\,27\big(l(l+1)t^{3}+2l(m-2)t^{2}-2(m-1)(l+1)t\nonumber\\ \displaystyle & & \displaystyle \quad \qquad -\,(m-1)(m-2)\big)/(2t^{2})\Big)\nonumber\\ \displaystyle R_{6} & := & \displaystyle L_{5}=\Big(-3\big(3(l-1)t^{2}-(l-2)(m-2)t+3(m-1)\big)/t,\nonumber\\ \displaystyle & & \displaystyle \quad \qquad 27\big((l-1)(l-2)t^{3}+2(l-1)(m-2)t^{2}\nonumber\\ \displaystyle & & \displaystyle \quad \qquad +\,2(l-2)(m-1)t+(m-1)(m-2)\big)/(2t^{2})\Big).\nonumber\end{eqnarray}$$

7.5 $F^{(6)}(\overline{k}(t_{6}))$ in the generic case

Let $Q_{1},\ldots ,Q_{8}$ be the basis of the Mordell–Weil group for $F_{E_{u},E_{v}}^{(3)}$ described in Theorem 4.8. By abuse of notation, let $Q_{1},\ldots ,Q_{8}$ be their base change (pullback) to $F_{E_{r,f},E_{q,g}}^{(6)}$ defined by (7.3), by the map

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706} & = & \displaystyle (r+1)(r-3)^{3}/\big((r-1)(r+3)^{3}\big)\qquad \text{and}\nonumber\\ \displaystyle \unicode[STIX]{x1D707} & = & \displaystyle (q+1)(q-3)^{3}/\big((q-1)(q+3)^{3}\big).\nonumber\end{eqnarray}$$

Similarly, let $R_{1},\ldots ,R_{8}$ be the base change of the basis of $^{t_{3}}F_{E_{\unicode[STIX]{x1D706}},E_{\unicode[STIX]{x1D707}}}^{(3)}(\bar{k}(t_{3}))$ . ( $R_{7}$ and $R_{8}$ are shown only in the auxiliary files.) Define

$$\begin{eqnarray}\displaystyle S_{1} & = & \displaystyle (Q_{2}+Q_{3}+Q_{6}+Q_{8}+R_{1})/2\nonumber\\ \displaystyle S_{2} & = & \displaystyle (Q_{1}+Q_{3}+Q_{7}+Q_{8}-R_{2})/2\nonumber\\ \displaystyle S_{3} & = & \displaystyle (Q_{1}+Q_{3}+R_{1}-R_{7}+R_{8})/2\nonumber\\ \displaystyle S_{4} & = & \displaystyle (Q_{2}+Q_{3}+R_{4}+R_{5}-R_{8})/2.\nonumber\end{eqnarray}$$

These are sections of $F_{E_{r,f},E_{q,g}}^{(6)}$ , that is, the expressions in parentheses are (uniquely) divisible by $2$ in the Mordell–Weil group. Explicit formulas are also given in auxiliary files.

8.1 Class number 1 case

It is well known that there are thirteen discriminants of class number  $1$ .

$$\begin{eqnarray}\displaystyle h_{D} & = & \displaystyle 1\Longleftrightarrow D\nonumber\\ \displaystyle & = & \displaystyle -3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163.\nonumber\end{eqnarray}$$

(See for example [Reference CoxCo2, Theorem 7.30] and its references.) From these we see that only the fields $\mathbb{Q}(\sqrt{-1})$ , $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(\sqrt{-7})$ possess nonmaximal order of class number  $1$ . In the following table, $D$ is a discriminant with $h_{D}=1$ , $f$ is the conductor of the order, $\unicode[STIX]{x1D70F}$ is the root of the quadratic form, and $E$ is an example of $E$ having $j(\unicode[STIX]{x1D70F})$ as its $j$ -invariant.

Table 1. Discriminants with class number 2.

Table 2. $\mathbb{Q}$ -curves over quadratic fields and associated $K3$ surfaces.

8.2 Class number 2 case

It is also known that there are only finitely many negative discriminants $D$ whose class number equals  $2$ (see for example [Reference Ireland and RosenIR, pp. 358–361]). Table 1 shows twenty nine such $D$ , together with $j(\unicode[STIX]{x1D70F}_{1})$ . Table 2 shows an example of $E_{1}$ for each  $j(\unicode[STIX]{x1D70F}_{1})$ , together with $F_{E_{1},E_{2}}^{(n)}$ where $E_{2}$ is the Galois conjugate of $E_{1}$ Footnote ³ . In these equations $X$ and $Y$ are rescaled so that the coefficients become simpler. However, $t$ is the original parameter $t_{n}$ , and thus some equations contain a variable $\unicode[STIX]{x1D700}$ , which indicates the fundamental unit of the real quadratic field $\mathbb{Q}(j(\unicode[STIX]{x1D70F}_{1}))$ . By rescaling the elliptic parameter $t$ suitably as in Lemma 8.3, we obtain elliptic $K3$ surfaces defined over  $\mathbb{Q}$ .

Remark 8.8. As remarked earlier, the equation of the Inose surface may be given in the form

$$\begin{eqnarray}Y^{2}=X^{3}-3\sqrt[3]{J_{1}J_{2}}\,X+t+\frac{1}{t}-2\sqrt{(1-J_{1})(1-J_{2})},\end{eqnarray}$$

where $J_{i}=j(\unicode[STIX]{x1D70F}_{i})/1728$ . Quite often $\sqrt[3]{j(\unicode[STIX]{x1D70F}_{1})j(\unicode[STIX]{x1D70F}_{2})}$ becomes a rational integer (see [Reference CoxCo2, Section 12]), and so does $\sqrt{(1728-j(\unicode[STIX]{x1D70F}_{1}))(1728-j(\unicode[STIX]{x1D70F}_{2}))}$ . In that case the above equation gives a model of the Inose surface over  $\mathbb{Q}$ . However, since that is not always the case, we uniformly started from (2.3). By doing so, we can also keep track of the field of definition of $F_{E_{1},E_{2}}^{(n)}(\overline{\mathbb{Q}}(t_{n}))$ .