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 $\mathbf{Q}$.

Keywords: Hessian elliptic curves, twists of elliptic curves, cubic fields

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@article{AMBP_2009__16_1_27_0, author = {Miyake, Katsuya}, title = {Twists of {Hessian} {Elliptic} {Curves} and {Cubic} {Fields}}, journal = {Annales math\'ematiques Blaise Pascal}, pages = {27--45}, publisher = {Annales math\'ematiques Blaise Pascal}, volume = {16}, number = {1}, year = {2009}, doi = {10.5802/ambp.251}, zbl = {1182.11026}, mrnumber = {2514525}, language = {en}, url = {http://www.numdam.org/articles/10.5802/ambp.251/} }

TY - JOUR AU - Miyake, Katsuya TI - Twists of Hessian Elliptic Curves and Cubic Fields JO - Annales mathématiques Blaise Pascal PY - 2009 SP - 27 EP - 45 VL - 16 IS - 1 PB - Annales mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.251/ DO - 10.5802/ambp.251 LA - en ID - AMBP_2009__16_1_27_0 ER -

Miyake, Katsuya. Twists of Hessian Elliptic Curves and Cubic Fields. Annales mathématiques Blaise Pascal, Volume 16 (2009) no. 1, pp. 27-45. doi : 10.5802/ambp.251. http://www.numdam.org/articles/10.5802/ambp.251/

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