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 $1/7$

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