## The modular curve $X_{37}$

Curve name $X_{37}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{9}$
Meaning/Special name
Chosen covering $X_{9}$
Curves that $X_{37}$ minimally covers $X_{9}$, $X_{17}$, $X_{18}$
Curves that minimally cover $X_{37}$ $X_{123}$, $X_{37a}$, $X_{37b}$, $X_{37c}$, $X_{37d}$
Curves that minimally cover $X_{37}$ and have infinitely many rational points. $X_{123}$, $X_{37a}$, $X_{37b}$, $X_{37c}$, $X_{37d}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{37}) = \mathbb{Q}(f_{37}), f_{9} = \frac{8f_{37}^{2} - 1}{f_{37}^{2} + f_{37} + \frac{1}{8}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 201x + 199$, with conductor $4480$
Generic density of odd order reductions $2659/10752$