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Let $E$ be an elliptic curve over a number field $F$. As usual, let

$\bar{\rho}:\mathrm{Gal}(\overline{F}/F)\to\mathrm{GL}(E[p])$ be the mod $p$ Galois representation associated to $E$. It is known that $\bar{\rho}$ is unramified at primes of good reduction. Let $\ell$ be a prime which $\bar{\rho}$ ramifies. Can we compute the ramification index for $\ell$ explicitly?

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  • $\begingroup$ I think the unramified statement is only true for primes of good reduction not lying over $p$. For $\ell$ a prime of bad reduction not lying over $2$ or $3$, it's pretty easy to do in terms of the Neron model. $\endgroup$
    – Will Sawin
    Apr 29, 2017 at 16:53
  • $\begingroup$ @WillSawin Maybe, I should rephrase the question: I am finding abstract formula for ramification indices. For example, I want to know whether there is something like: if $y^2=x^3-Ax-B$, can we write down the ramification index $e_\ell$ for $\ell$ in terms of $A$ and $B$? $\endgroup$
    – User0829
    Apr 29, 2017 at 17:42
  • $\begingroup$ I think you can, and the first step would be Tate's algorithm to find the fiber type, and then for $\ell$ and $p$ at least $5$ or so there should be a relatively simple formula for the ramification index in terms of the fiber type. $\endgroup$
    – Will Sawin
    Apr 29, 2017 at 18:29

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