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Let $E$ be an elliptic curve over a number field $K$ and $p$ a prime. Suppose that $E$ has a $K$-rational $p$-torsion, which gives the short exact sequence $0\to\mathbb{Z}/p\to E[p]\to\mu_p\to0$ of Galois modules. Let us assume that this sequence splits, i.e. $E[p]\simeq\mathbb{Z}/p\times\mu_p$ as Galois modules. Set $E':=E/\mu_p$. Now, my questions is:

Is the set of bad primes for the elliptic curve $E'$ the same as the set of bad primes for $E$?

I am also wondering whether there is a general theory in this line, for example, how to compute the set of bad rimes for the quotient elliptic curve $E/H$ for "Galois stable" subgroup $H$ of $E$.

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    $\begingroup$ E and E’ are isogenous, so their places of bad reduction are the same by Neron Ogg Shafarevich $\endgroup$ Aug 17, 2023 at 15:33

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As has already been remarked, but more generally, if $K$ is a number field and $A/K$ and $B/K$ are abelian varieties that are isogenous over $K$, then the criterion of Neron-Ogg-Shafarevich has as a quick corollary that $A$ and $B$ have the same set of primes of bad reduction. The paper with the proof is by Serre and Tate. (Serre, Jean-Pierre; Tate, J., Good reduction of abelian varieties, Ann. Math. (2) 88, 492-517 (1968). ZBL0172.46101.) If you want a proof just for elliptic curves, you could look at Section VII.7 in (Silverman, Joseph H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106. New York, NY: Springer (ISBN 978-0-387-09493-9/hbk; 978-0-387-09494-6/ebook). xx, 513 p. (2009). ZBL1194.11005).

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