## The modular curve $X_{97}$

Curve name $X_{97}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 3 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 6 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{39}$
Curves that $X_{97}$ minimally covers $X_{39}$, $X_{45}$, $X_{47}$
Curves that minimally cover $X_{97}$ $X_{254}$, $X_{260}$, $X_{264}$, $X_{275}$, $X_{301}$, $X_{303}$, $X_{304}$, $X_{308}$, $X_{309}$, $X_{310}$, $X_{311}$, $X_{312}$, $X_{377}$, $X_{380}$
Curves that minimally cover $X_{97}$ and have infinitely many rational points. $X_{304}$, $X_{308}$, $X_{309}$, $X_{312}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{97}) = \mathbb{Q}(f_{97}), f_{39} = \frac{4f_{97}^{2} + 1}{f_{97}^{2} + f_{97} - \frac{1}{4}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 558481500x - 5079892400000$, with conductor $1159200$
Generic density of odd order reductions $1343/5376$