## The modular curve $X_{8}$

 Curve name $X_{8}$ Index $6$ Level $2$ Genus $0$ Does the subgroup contain $-I$? Yes Generating matrices Images in lower levels Meaning/Special name Elliptic curves with full $2$-torsion over $\mathbb{Q}$ Chosen covering $X_{6}$ Curves that $X_{8}$ minimally covers $X_{2}$, $X_{6}$ Curves that minimally cover $X_{8}$ $X_{24}$, $X_{25}$, $X_{38}$, $X_{46}$, $X_{8a}$, $X_{8b}$, $X_{8c}$, $X_{8d}$ Curves that minimally cover $X_{8}$ and have infinitely many rational points. $X_{24}$, $X_{25}$, $X_{38}$, $X_{46}$, $X_{8a}$, $X_{8b}$, $X_{8c}$, $X_{8d}$ Model $\mathbb{P}^{1}, \mathbb{Q}(X_{8}) = \mathbb{Q}(f_{8}), f_{6} = \frac{8f_{8}^{2} + 24}{f_{8} - 1}$ Info about rational points None Comments on finding rational points None Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 68x + 182$, with conductor $315$ Generic density of odd order reductions $1/7$