Research proof index

The rational points of the quadratic twist, viewed inside E(L) via the isomorphism over L, are exactly the points of E(L) on which the nontrivial element of Gal(L/K) acts as -1 (just as E(K) consists of the points on which it acts as +1). One inclusion is Affine.Point.map_baseChange (the base change of a K-point is σ-fixed) together wi...

An explicit L-isomorphism (Eᶿ)ᴸ ≅ Eᴸ (the change of variables of the module docstring) which moreover is anti-equivariant for the Galois action: its conjugate by the nontrivial σ ∈ Gal(L/K) differs from it by the automorphism [-1] of E. This nontrivial cocycle is the origin of the twist being a nontrivial form of E.

Classification of the forms of E split by L/K, for j(E) ∉ {0, 1728}: an elliptic curve over K which becomes isomorphic to E over L is isomorphic over K either to E or to its quadratic twist by L. (Such forms are classified by H¹(Gal(L/K), Aut(E_L)) = Hom(ℤ/2, {±1}), which has order 2. For j ∈ {0, 1728} the automorphism group is large...

The quadratic twist becomes isomorphic to E after base change to L. (Over a field, isomorphisms of Weierstrass curves are exactly the admissible changes of variables WeierstrassCurve.VariableChange, acting via •.) The point-level consequences of this isomorphism, which is what most applications need, are recorded separately in `quadraticTwistPoint...

If j(E) ∉ {0, 1728} (so that the only automorphisms of E are ±1) then the quadratic twist is not isomorphic to E over K: twisting by L/K is a nontrivial operation. This can fail for j ∈ {0, 1728}: e.g. for E : y² = x³ + x of j-invariant 1728 over K = ℚ(i), the quadratic twist by any L = K(d^{1/2}) with d ∈ (K^×)⁴ ∖ (K^×)² is is...

The classical formula for the quadratic twist away from characteristic 2. Suppose char K ≠ 2, so that after completing the square we may assume E has the form y² = x³ + a₂x² + a₄x + a₆, and suppose L = K(α) where α² = d is a nonsquare in K (every separable quadratic extension arises this way when char K ≠ 2). Then the quadratic twist of E...