TITLE: Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$
QUESTION [8 upvotes]: Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ and $f_i$ the eigencuspform of weight $2$ attached to $E_i$. Express $f_1=\sum a_i q^i$ and $f_2=\sum b_i q^i$.
Suppose that the residual Galois representations $E_1[p]\simeq E_2[p]$. Note that $E_1$ and $E_2$ have bad reduction at the same primes.
It is easy to see that $a_l\equiv b_l\mod{p}$ at a prime $l$ of good reduction for both elliptic curves (since the residual representations are unramified at $l$). Does it follow that $a_l\equiv b_l\mod{p}$ at a prime $l$ of bad reduction for both elliptic curves. If so how do we show this? Perhaps this comes down to showing that $E_1$ has the same kind of bad reduction at $p$ as $E_2$. I know that if $E_1$ has additive reduction then so does $E_2$, but I'm not sure about distinguishing between split multiplicative and non-split multiplicative reduction.

REPLY [3 votes]: Note that a conjecture attributed to Frey and Mazur says that for each elliptic curve $E_1$, there exists a constant $C$ depending on $E_1$, such that for every prime $p>C$ and for every elliptic curve $E_2$ defined over $\mathbb{Q}$ with $E_1[p]\simeq E_2[p]$ as Galois modules, one has $E_1$ isogenous to $E_2$. In particular, $E_1$ and $E_2$ have the same $L$-functions and hence the same coefficients $a_\ell$ (including at the bad primes).
It is even conjectured that $C$ can be taken independently of $E_1$. Moreover there is no known pair of non-isogenous elliptic curves $(E_1,E_2)$ such that $E_1[p]\simeq E_2[p]$ for some prime $p>17$. 
If you want to learn more about this conjecture you can have a look at the recent preprint by Cremona and Freitas "Global methods for the symplectic type of congruences between elliptic curves": https://arxiv.org/abs/1910.12290