Pairs of mod~3 and mod~5 representations arising from elliptic curves

For an elliptic curve~$E_0$ over~$\mathbb{Q}$, we study the mod~$3$ representations, restricted to a decomposition group at $3$, of elliptic curves over~$\mathbb{Q}$ whose mod~$5$ representations are equivalent to the mod~$5$ representation given by~$E_0$. Complete lists are obtained describing the mod~$3$ representations up to equivalence and twist by the quadratic unramified character of~$\mathbb{Q}_3.$

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Published 1 January 1999