Step 5: The function field of Ef
We want to find two functions h1, h2 on Ef whose zeros and poles are
contained in S, and which generate the function field of Ef .
If |S| ≤ 2, this is impossible.
Otherwise:
1. Generate principal divisors on E supported in S (this is possible since
S consists of torsion points, by the Manin-Drinfeld theorem).
2. Take two such divisors D1, D2 and compute functions h1, h2 ∈ Q(E)
with these divisors.
3. Compute the minimal polynomial P ∈ Q[X1, X2] of (h1, h2).
4. Check the partial degrees of P to decide whether Q(Ef ) = Q(h1, h2).
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