Section: New Results

Number and function fields

The article [13] written by J. Brau and J. Nathan on “Elliptic curves with 2-torsion contained in the 3-torsion field” has been published. This article study the modular curve $X^{\text{'}}\left(6\right)$ of level 6 defined over $\mathbb{Q}$ whose $\mathbb{Q}$-rational points correspond to $j$-invariants of elliptic curves $E$ over $\mathbb{Q}$ for which $\mathbb{Q}\left(E\right[2\left]\right)$ is a subfield of $\mathbb{Q}\left(E\right[3\left]\right)$. The authors characterize the $j$-invariants of elliptic curves with this property by exhibiting an explicit model of $X^{\text{'}}\left(6\right)$. $X^{\text{'}}\left(6\right)\left(\mathbb{Q}\right)$ then gives an infinite family of examples of elliptic curves with non-abelian "entanglement fields," which is relevant to the systematic study of correction factors of various conjectural constants for elliptic curves over $\mathbb{Q}$.