## The modular curve $X_{188}$

Curve name $X_{188}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{98}$
Curves that $X_{188}$ minimally covers $X_{98}$, $X_{100}$, $X_{101}$
Curves that minimally cover $X_{188}$ $X_{449}$, $X_{450}$, $X_{455}$, $X_{462}$, $X_{188a}$, $X_{188b}$, $X_{188c}$, $X_{188d}$, $X_{188e}$, $X_{188f}$, $X_{188g}$, $X_{188h}$
Curves that minimally cover $X_{188}$ and have infinitely many rational points. $X_{188a}$, $X_{188b}$, $X_{188c}$, $X_{188d}$, $X_{188e}$, $X_{188f}$, $X_{188g}$, $X_{188h}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{188}) = \mathbb{Q}(f_{188}), f_{98} = \frac{f_{188}^{2} + 2}{f_{188}^{2} - 2}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 756x - 7808$, with conductor $126$
Generic density of odd order reductions $193/1792$