Twists of Hessian Elliptic Curves and Cubic Fields

Miyake, Katsuya

Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 27-45.

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

Classification : 11G05, 12F05
Mots clés : Hessian elliptic curves, twists of elliptic curves, cubic fields

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