Twists of Hessian Elliptic Curves and Cubic Fields
Annales mathématiques Blaise Pascal, Volume 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 $\stackrel{˜}{H}\left(\mu ,t\right),\mu ,t\in \mathbf{Q}$ over the Galois closures of cubic fields. We also show that ${H}_{\mu }$ is a twist of $\stackrel{˜}{H}\left(\mu ,t\right)$ over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, $R\left(t;X\right):={X}^{3}+tX+t,t\in \mathbf{Q}-\left\{0,-27/4\right\}$, to parametrize all of quadratic fields and cubic ones. It should be noted that $\stackrel{˜}{H}\left(\mu ,t\right)$ 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 $\stackrel{˜}{H}\left(\mu ,t\right)$ by a certain subset of the cubic field. In the case of $\mu =0$, we give a criterion for $\stackrel{˜}{H}\left(0,t\right)$ to have a rational point over $\mathbf{Q}$.

DOI: 10.5802/ambp.251
Classification: 11G05,  12F05
Keywords: Hessian elliptic curves, twists of elliptic curves, cubic fields
Miyake, Katsuya 1

1 Department of Mathematics School of Fundamental Science and Engineering Waseda University 3–4–1 Ohkubo Shinjuku-ku Tokyo, 169-8555 Japan
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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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