Classify all elliptic curves with Galois groups of some form?
Classify l-adic images of Galois.

- Find subgroups $H$that occur as the image of Galois
- Compute equations for modular curves $X_H$
- Determine rational $j$ invariants  $j_H: X_H \to X(1) \cong \PP^1_{/\QQ}$
- Provably find all rational points on each $X_H$.

Sporadic: not cuspidal and not [complex multiplication](complex%20multiplication)
Compute genus using [Riemann-Hurwitz](Riemann-Hurwitz).