## The modular curve $X_{90}$

Curve name $X_{90}$
Index $24$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 1 & 3 \\ 6 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 1 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{11}$
Meaning/Special name
Chosen covering $X_{28}$
Curves that $X_{90}$ minimally covers $X_{28}$, $X_{39}$, $X_{50}$
Curves that minimally cover $X_{90}$ $X_{256}$, $X_{262}$, $X_{280}$, $X_{281}$, $X_{282}$, $X_{283}$, $X_{284}$, $X_{285}$
Curves that minimally cover $X_{90}$ and have infinitely many rational points. $X_{281}$, $X_{284}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{90}) = \mathbb{Q}(f_{90}), f_{28} = \frac{2f_{90}^{2} - 4}{f_{90}^{2} + 4f_{90} + 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 3371423x + 801711878$, with conductor $16940$
Generic density of odd order reductions $1427/5376$