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.

Classification : 11G05,  12F05
Mots clés : Hessian elliptic curves, twists of elliptic curves, cubic fields
     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},
     mrnumber = {2514525},
     zbl = {1182.11026},
     language = {en},
     url = {}
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.

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