Given an elliptic curve $E$ defined over $\mathbb{Z}$, and a prime $p$, I know that Hasse's theorem gives, *when $p$ is a good prime*, a relation between the number of solutions over $\mathbb{F}\_{p^n}$ and the number of solutions over 
$\mathbb{F}\_p$ (for this, the coefficients of the equation are reduced mod $p$). 


Is there such a relation also at the bad primes?



How many points are there on an elliptic curve reduced at a bad prime?
If the reduction is additive, there are $p+1$ points including one singular point. If it is split multiplicative it is $p$ and if non-split multiplicative, then it is $p+2$. See Washington "Elliptic curves, Number Theory and Cryptography ", section 2.10 on page 59.


... and see François comment below for $n>1$.


