## The modular curve $X_{32}$

Curve name $X_{32}$
Index $12$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{13}$
Curves that $X_{32}$ minimally covers $X_{13}$
Curves that minimally cover $X_{32}$ $X_{79}$, $X_{96}$, $X_{115}$, $X_{116}$, $X_{157}$, $X_{158}$, $X_{32a}$, $X_{32b}$, $X_{32c}$, $X_{32d}$, $X_{32e}$, $X_{32f}$, $X_{32g}$, $X_{32h}$, $X_{32i}$, $X_{32j}$, $X_{32k}$, $X_{32l}$
Curves that minimally cover $X_{32}$ and have infinitely many rational points. $X_{79}$, $X_{96}$, $X_{115}$, $X_{116}$, $X_{32a}$, $X_{32b}$, $X_{32c}$, $X_{32d}$, $X_{32e}$, $X_{32f}$, $X_{32g}$, $X_{32h}$, $X_{32i}$, $X_{32j}$, $X_{32k}$, $X_{32l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{32}) = \mathbb{Q}(f_{32}), f_{13} = -f_{32}^{2} - 8$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 195x - 1000$, with conductor $1845$
Generic density of odd order reductions $9/56$