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On the modular curves $Y_E(7)$

Authors: Emmanuel Halberstadt and Alain Kraus
Journal: Math. Comp. 69 (2000), 1193-1206
MSC (1991): Primary 11Gxx
Published electronically: May 21, 1999
MathSciNet review: 1651758
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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 Weierstraß equation 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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Additional Information

Emmanuel Halberstadt
Affiliation: Université Paris VI, Laboratoire de Mathématiques Fondamentales, UFR 921, 4, place Jussieu, 75252 Paris Cedex 05, France

Alain Kraus
Affiliation: Université Paris VI, Institut de Mathématiques, Case 247, 4, place Jussieu, 75252 Paris Cedex 05, France

Received by editor(s): August 8, 1997
Received by editor(s) in revised form: July 24, 1998
Published electronically: May 21, 1999
Article copyright: © Copyright 2000 American Mathematical Society