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:



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



Primes of bad reductions for quotients of elliptic curves
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**](https://doi.org/10.2307/1970722), Ann. Math. (2) 88, 492-517 (1968). [ZBL0172.46101](https://zbmath.org/?q=an:0172.46101).) 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**](https://doi.org/10.1007/978-0-387-09494-6), 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](https://zbmath.org/?q=an:1194.11005)).


