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.



Equivalent notions of congruence for elliptic curves over $\mathbb{Q}$
I found myself wondering the same thing a couple of weeks ago. Even with the restriction that $p > 2$ and the residual representation is irreducible, it does not follow that $E\_{1}$ and $E\_{2}$ have the same primes of bad reduction, nor that $a\_{\ell}(E\_{1}) \equiv a\_{\ell}(E\_{2}) \pmod{p}$ for primes of bad reduction. 


For example, take $p = 5$ and $E\_{1} : y^{2} + xy + y = x^{3} + 4x - 6$ (aka $X\_{0}(14)$). The mod $5$ Galois representation attached to $E\_{1}$ is surjective. For any elliptic curve $E$, Rubin and Silverberg write down an isomorphism $X\_{E}(5) \cong \mathbb{P}^{1}$, the modular curve parametrizing elliptic curves $F$ so that $F[5]$ and $E[5]$ are isomorphic via and isomorphism respecting the Weil pairing. For $E = E\_{1}$, one can take $E\_{2} : y^{2} + xy = x^{3} - x^{2} - 4492x + 126416$. This curve has $E\_{1}[5] \cong E\_{2}[5]$, but $E\_{2}$ has bad reduction at $5$ and $E\_{1}$ has good reduction at $5$. It is not surprising that one can fail to have $a\_{\ell}(E\_{1}) \equiv a\_{\ell}(E\_{2}) \pmod{p}$ when one of $E\_{1}$ and $E\_{2}$ has good reduction at $\ell$ and the other doesn't.


One can also have $a\_{\ell}(E\_{1}) \not\equiv a\_{\ell}(E\_{2}) \pmod{p}$ when both curves have bad reduction at $\ell$. Looking in the same family (of curves directly $5$-congruent to $E\_{1}$) one can find curves $E\_{2}$ and $E\_{3}$ that have multiplicative reduction at $199$ where one curve has $a\_{199}(E\_{2}) = -1$ and the other has $a\_{199}(E\_{3}) = 1$. The examples I found are a bit horrendous:
$$ E\_{2} : y^2 + xy = x^3 - x^2 - 
183192520591565859491828595856323163822228663229886627562x + 
1062123179152064591727007023640066384957014639073044998393946451273297951607053468340$$
and
$$ E\_{3} : y^2 + xy + y = x^3 - 
187739104428369548177010660620161617529156118721103898180850571x + 
1108087782673654453656602976287807709340970590450649179565293267500311360376488672021879300982. $$


