## The modular curve $X_{91}$

Curve name $X_{91}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 2 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
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 $X_{91}$ minimally covers $X_{39}$, $X_{43}$, $X_{50}$
Curves that minimally cover $X_{91}$ $X_{256}$, $X_{259}$, $X_{261}$, $X_{276}$, $X_{286}$, $X_{287}$, $X_{288}$, $X_{289}$, $X_{290}$, $X_{291}$, $X_{300}$, $X_{352}$
Curves that minimally cover $X_{91}$ and have infinitely many rational points. $X_{288}$, $X_{289}$, $X_{291}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{91}) = \mathbb{Q}(f_{91}), f_{39} = \frac{f_{91}^{2} + 2}{f_{91}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 8786x - 296215$, with conductor $11109$
Generic density of odd order reductions $401/1792$