Let $\mathcalE$ be an elliptic curve, $m$ a positive number and $\E[m]$ the $m$-torsion subgroup of $\mathcalE$. Let $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ form a basis of $\E[m]$. We prove that $\mathbbQ(\E[m])=\mathbbQ(x_1\,,x_2\,,\zeta_m\,,y_1)$ in general. For the case $m=3$ we provide a description of all the possible extensions $\mathbbQ(\E[3])$ in terms of generators, degree and Galois groups.
Number fields generated by the 3-torsion points of an elliptic curve
A. Bandini;
2012-01-01
Abstract
Let $\mathcalE$ be an elliptic curve, $m$ a positive number and $\E[m]$ the $m$-torsion subgroup of $\mathcalE$. Let $P_1=(x_1,y_1)$, $P_2=(x_2,y_2)$ form a basis of $\E[m]$. We prove that $\mathbbQ(\E[m])=\mathbbQ(x_1\,,x_2\,,\zeta_m\,,y_1)$ in general. For the case $m=3$ we provide a description of all the possible extensions $\mathbbQ(\E[3])$ in terms of generators, degree and Galois groups.File in questo prodotto:
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