## The modular curve $X_{22}$

Curve name $X_{22}$
Index $8$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 2 & 3 \\ 1 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 7 \\ 6 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $1$ $X_{1}$ $4$ $4$ $X_{7}$
Meaning/Special name Elliptic curves whose discriminant is minus twice a square with $a_{p}(E) \equiv 0 \pmod{4}$ for $p \equiv 5 \text{ or } 7 \pmod{8}$
Chosen covering $X_{7}$
Curves that $X_{22}$ minimally covers $X_{4}$, $X_{7}$
Curves that minimally cover $X_{22}$ $X_{56}$, $X_{57}$, $X_{83}$, $X_{179}$
Curves that minimally cover $X_{22}$ and have infinitely many rational points. $X_{56}$, $X_{57}$, $X_{83}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{22}) = \mathbb{Q}(f_{22}), f_{7} = \frac{\frac{1}{6}f_{22}^{2} - 3}{f_{22}^{2} + 12f_{22} + 30}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 88333x + 10222037$, with conductor $44800$
Generic density of odd order reductions $955/1792$