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 \(j_H: X_H \to X(1) \cong {\mathbf{P}}^1_{/{\mathbf{Q}}}\)
- Provably find all rational points on each \(X_H\).
Sporadic: not cuspidal and not complex multiplication Compute genus using Riemann-Hurwitz.