On the modular curves 𝑌_{𝐸}(7)

Halberstadt, Emmanuel; Kraus, Alain

doi:10.1090/S0025-5718-99-01123-0

On the modular curves $Y_E(7)$
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by Emmanuel Halberstadt and Alain Kraus;

Math. Comp. 69 (2000), 1193-1206

DOI: https://doi.org/10.1090/S0025-5718-99-01123-0

Published electronically: May 21, 1999

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Abstract:

Let $E$ denote an elliptic curve over $\mathbf {Q}$ and $Y_E(7)$ the modular curve classifying the elliptic curves $E’$ over $\mathbf {Q}$ such that the representations of $\operatorname {Gal}(\overline {\mathbf Q}/\mathbf {Q})$ in the 7-torsion points of $E$ and of $E’$ are symplectically isomorphic. In case $E$ is given by a Weierstraßequation such that the $c_4$ invariant is a square, we exhibit here nontrivial points of $Y_E(7)(\mathbf {Q})$. From this we deduce an infinite family of curves $E$ for which $Y_E(7)(\mathbf {Q})$ has at least four nontrivial points.

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