Step 3: Images of cusps
Recall that ϕ: X1(N) → Ef is given by τ 7→ [
R τ
0
ωf ].
Hypothesis: ϕ is defined over Q(ζN ).
(This does not always hold.)
1. Enumerate the cusps c1, . . . , cs of X1(N).
2. For each 1 ≤ i ≤ s, compute zi =
R ci
0
ωf .
3. Compute pi,C = ellztopoint(Ef , zi ) ∈ Ef (C).
4. Writing pi,C = (xi , yi ), reconstruct xi , yi in Q(ζN ) using lindep.
5. Check whether pi = (xi , yi ) ∈ Ef (Q(ζN )).
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