# Isogeny classes and elliptic curves over finite fields

Fix a conductor and a prime $p$. Then

1) Do the elliptic curves in the same isogeny class after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$

2) Do the elliptic curves belonging to two different isogeny classes corresponding to the fixed conductor, after reduction modulo $p$ have the same number of points over the finite field $\mathbb{F}_{p} ?$

• I'm a little confused by your answer to the second question, which seems to assume that the reduction is bad. From the way the question is written I would instead have assumed, if anything, that the reduction is good (i.e., $p$ doesn't divide the conductor). But it's a little vague... – Pete L. Clark Dec 4 '13 at 6:44
If in (2) you are asking whether the conductor entirely determines the number of points on all reductions, the answer is most assuredly not. If that were the case then there would only be one cusp form over $\mathbf{Q}$ for each $\Gamma_0(N)$ where $N$ is your conductor. But there are counterexamples all over the place for this.
For your specific example, take $p =3$ and $N=37$. The 37a isogeny class is supersingular while the 37b isogeny class is not. Therefore the number of points mod $3$ on say 37.a1 and 37.b1 (LMFDB labels) will be different.