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WARNING: I haven't exactly tried the problems below. I don't know how computationally difficult they are for sure.

**Problem 1:** Let $\rho_{E,2}$ denote the mod 2 representation attached to $E$ for each of the following elliptic curves: 17a1, 32a1, 32a2, 37a1. For each, compute explicitly the matrix of $\rho_{E,2}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13. Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.

**Problem 2:** Let $\rho_{E,4}$ denote the mod 4 representation attached to $E$ for the curve 32a1. This is the homomorphism $G_\QQ\to\text{GL}_2(\ZZ/4\ZZ)$ defined by the action of $G_\QQ$ on $E[4]$. Be sure to check that $X^2 - a_p(E)X + p$ is the charpoly of $\rho_{E,2}(\text{Frob}_P)$.

Try to compute explicitly the matrix of $\rho_{E,4}(\text{Frob}_P)$, where $P$ is a choice of prime ideal over each of 3,5,7,11,13.