## The modular curve $X_{190}$

Curve name $X_{190}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{99}$
Curves that $X_{190}$ minimally covers $X_{99}$, $X_{100}$, $X_{101}$
Curves that minimally cover $X_{190}$ $X_{446}$, $X_{449}$, $X_{462}$, $X_{463}$, $X_{190a}$, $X_{190b}$, $X_{190c}$, $X_{190d}$, $X_{190e}$, $X_{190f}$, $X_{190g}$, $X_{190h}$
Curves that minimally cover $X_{190}$ and have infinitely many rational points. $X_{190a}$, $X_{190b}$, $X_{190c}$, $X_{190d}$, $X_{190e}$, $X_{190f}$, $X_{190g}$, $X_{190h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{190}) = \mathbb{Q}(f_{190}), f_{99} = \frac{f_{190}^{2} - 8}{f_{190}^{2} + 8f_{190} + 8}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 108x + 288$, with conductor $600$
Generic density of odd order reductions $635/5376$