## The modular curve $X_{96}$

Curve name $X_{96}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{25}$
Curves that $X_{96}$ minimally covers $X_{25}$, $X_{32}$, $X_{36}$
Curves that minimally cover $X_{96}$ $X_{184}$, $X_{185}$, $X_{192}$, $X_{193}$, $X_{194}$, $X_{206}$, $X_{208}$, $X_{215}$, $X_{246}$, $X_{249}$, $X_{313}$, $X_{314}$, $X_{315}$, $X_{316}$, $X_{386}$, $X_{404}$, $X_{96a}$, $X_{96b}$, $X_{96c}$, $X_{96d}$, $X_{96e}$, $X_{96f}$, $X_{96g}$, $X_{96h}$, $X_{96i}$, $X_{96j}$, $X_{96k}$, $X_{96l}$, $X_{96m}$, $X_{96n}$, $X_{96o}$, $X_{96p}$, $X_{96q}$, $X_{96r}$, $X_{96s}$, $X_{96t}$
Curves that minimally cover $X_{96}$ and have infinitely many rational points. $X_{185}$, $X_{192}$, $X_{193}$, $X_{194}$, $X_{208}$, $X_{215}$, $X_{96a}$, $X_{96b}$, $X_{96c}$, $X_{96d}$, $X_{96e}$, $X_{96f}$, $X_{96g}$, $X_{96h}$, $X_{96i}$, $X_{96j}$, $X_{96k}$, $X_{96l}$, $X_{96m}$, $X_{96n}$, $X_{96o}$, $X_{96p}$, $X_{96q}$, $X_{96r}$, $X_{96s}$, $X_{96t}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{96}) = \mathbb{Q}(f_{96}), f_{25} = -f_{96}^{2}$
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 40148415x - 97904904944$, with conductor $6435$
Generic density of odd order reductions $19/168$