Given a lattice L \in C with fixed basis {w_1, w_2}, Let \delta(L) = | w_1 \bar{w_2} – w_2 \bar{w_1}|. Let f : C/L ---> C/L’ be an isogeny determined by multiplication by a \in C{0}. Via composition, f induces an inclusion M(L’) ---> M(L). show that:
- deg(f) = \delta(aL) / \delta(L’) = |a|^2 * \delta(L) / \delta(L’)
and
- The extension M(L)/M(L’) is Galois, with Gal(M(L)/M(L’)) isomorph to L/aL’.
