Twists of Hessian Elliptic Curves and Cubic Fields
Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 1, pp. 27-45.

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

DOI : 10.5802/ambp.251
Classification : 11G05, 12F05
Mots clés : 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, Tome 16 (2009) no. 1, pp. 27-45. doi : 10.5802/ambp.251. http://archive.numdam.org/articles/10.5802/ambp.251/

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