Let ${\mathcal{E}}$ be an elliptic curve, $m$ a positive number and $\E[m]$ the $m$-torsion subgroup of $\mathcal{E}$. Let $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ form a basis of $\E[m]$. We prove that $\mathbb{Q}(\E[m])=\mathbb{Q}(x_1\,,x_2\,,\zeta_m\,,y_1)$ in general. For the case $m=3$ we provide a description of all the possible extensions $\mathbb{Q}(\E[3])$ in terms of generators, degree and Galois groups.
Number fields generated by the 3-torsion points of an elliptic curve / Bandini, Andrea; L., Paladino. - In: MONATSHEFTE FÜR MATHEMATIK. - ISSN 0026-9255. - 168:2(2012), pp. 157-181. [10.1007/s00605-012-0377-x]
Number fields generated by the 3-torsion points of an elliptic curve
BANDINI, Andrea;
2012-01-01
Abstract
Let ${\mathcal{E}}$ be an elliptic curve, $m$ a positive number and $\E[m]$ the $m$-torsion subgroup of $\mathcal{E}$. Let $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ form a basis of $\E[m]$. We prove that $\mathbb{Q}(\E[m])=\mathbb{Q}(x_1\,,x_2\,,\zeta_m\,,y_1)$ in general. For the case $m=3$ we provide a description of all the possible extensions $\mathbb{Q}(\E[3])$ in terms of generators, degree and Galois groups.File in questo prodotto:
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