Q-curves
Definition
A Q-curve is an elliptic curve defined over Q which is isogenous to all
its Galois conjugates.
Example
Let K be a real quadratic field, and u ∈ K\{±1} such that 4u ∈ OK
and NK/Q(u) = 1. Then Ek : x + 1
x + y + 1
y + 4u = 0 is a Q-curve.
In this case the isogeny is defined over K.
Modularity theorem (Khare–Wintenberger, Ribet)
Let E be an elliptic curve over Q. Then E is a Q-curve if and only if
there exists a modular parametrisation ϕ: X1(N)Q → E.
Question. Can we make ϕ explicit?
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