import Mathlib

set_option autoImplicit false

open Polynomial

namespace WeierstrassCurve

variable {R : Type*} [CommRing R]

noncomputable def hasseInvariant (q : ℕ) (W : WeierstrassCurve R) : R :=
  (W.twoTorsionPolynomial.toPoly ^ ((q - 1) / 2)).coeff (q - 1)

def legendreCurve (t : R) : WeierstrassCurve R :=
  ⟨0, -(1 + t), 0, t, 0⟩

end WeierstrassCurve

namespace ModularCurve

def ssJSetHasse (q : ℕ) (K : Type*) [Field K] : Set K :=
  {j | ∀ (W : WeierstrassCurve K) [W.IsElliptic], W.j = j → W.hasseInvariant q = 0}

end ModularCurve