# Why do these elliptic curves of conductor $11$ have different representations modulo $5$?

Consider the elliptic curves $$E:y^2 + y = x^{3} - x^{2} - 7820 x - 263580,$$ $$E':y^2 + y = x^{3} - x^{2} - 10 x - 20,$$ and $$E'':y^2 + y = x^{3} - x^{2}$$ over $\mathbf{Q}$. All three of these curves have conductor $11$, and the space of newforms of weight $2$ for $\Gamma_0(11)$ has complex dimension $1$, so the modularity theorem tells us that their $\ell$-adic representations $$\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\operatorname{GL}_2(\mathbf{Q}_\ell)$$ are similar for all prime numbers $\ell$.

But according to the LMFDB if you reduce their $5$-adic representations modulo $5$ (after replacing the $5$-adic represention with one taking values in $\operatorname{GL}_2(\mathbf{Z}_5)$) they have different images, namely $E$ has $B.1.2$ , $E'$ has $Cs.1.2$, and $E''$ has $B.1.1$.

What's going on here?

• In order to define the residual representations, you first have to fix a $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$-stable lattice. In this case, the lattice is nothing more than the $\ell$-adic Tate module. Since $E, E',$ and $E''$ differ by cyclic isogenies of degree equal to powers of 5, the Galois actions on their 5-adic Tate modules differ. This does not contradict that their rational $\ell$-adic representations differ, as the rational $\ell$-adic Tate module $V_\ell(E) = T_\ell(E) \otimes \mathbf{Q}$ is invariant under isogeny. – Brandon Carter Mar 27 '18 at 18:44
• @BrandonCarter Ah that makes sense, so thank you! If you want to copy paste it as an answer I'll accept it. – RJM Mar 27 '18 at 18:48
• Related: Frey-Mazur conjecture. – Watson Apr 6 '18 at 18:23

In order to define the residual representations, you first have to fix a $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$-stable lattice. In this case, the lattice is nothing more than the $\ell$-adic Tate module. Since $E,E'$, and $E''$ differ by cyclic isogenies of degree equal to powers of 5, the Galois actions on their 5-adic Tate modules differ. This does not contradict that their rational $\ell$-adic representations differ, as the rational $\ell$-adic Tate module $V_\ell(E) = T_\ell(E) \otimes \mathbf{Q}$ is invariant under isogeny.