Twists of Hessian Elliptic Curves and Cubic Fields

In this paper we investigate Hesse’s elliptic curves Hμ : U + V 3 + W 3 = 3μUVW,μ ∈ Q − {1}, and construct their twists, Hμ,t over quadratic fields, and H̃(μ, t), μ, t ∈ Q over the Galois closures of cubic fields. We also show that Hμ is a twist of H̃(μ, t) over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, R(t;X) := X + tX + t, t ∈ Q − {0,−27/4}, to parametrize all of quadratic fields and cubic ones. It should be noted that H̃(μ, t) is a twist of Hμ as algebraic curves because it may not always have any rational points over Q. We also describe the set of Q-rational points of H̃(μ, t) by a certain subset of the cubic field. In the case of μ = 0, we give a criterion for H̃(0, t) to have a rational point over Q.