In this paper we investigate Hesse’s elliptic curves , and construct their twists, over quadratic fields, and over the Galois closures of cubic fields. We also show that is a twist of over the related cubic field when the quadratic field is contained in the Galois closure of the cubic field. We utilize a cubic polynomial, , to parametrize all of quadratic fields and cubic ones. It should be noted that is a twist of as algebraic curves because it may not always have any rational points over . We also describe the set of -rational points of by a certain subset of the cubic field. In the case of , we give a criterion for to have a rational point over .
@article{AMBP_2009__16_1_27_0, author = {Katsuya Miyake}, 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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/} }
TY - JOUR AU - Katsuya Miyake 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 - https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/ DO - 10.5802/ambp.251 LA - en ID - AMBP_2009__16_1_27_0 ER -
%0 Journal Article %A Katsuya Miyake %T Twists of Hessian Elliptic Curves and Cubic Fields %J Annales mathématiques Blaise Pascal %D 2009 %P 27-45 %V 16 %N 1 %I Annales mathématiques Blaise Pascal %U https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/ %R 10.5802/ambp.251 %G en %F AMBP_2009__16_1_27_0
Katsuya Miyake. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.251/
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