## The modular curve $X_{188h}$

Curve name $X_{188h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{188}$
Curves that $X_{188h}$ minimally covers
Curves that minimally cover $X_{188h}$
Curves that minimally cover $X_{188h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} + 1728t^{14} - 5184t^{12} - 6912t^{10} - 397440t^{8} - 27648t^{6} - 82944t^{4} + 110592t^{2} - 27648$ $B(t) = -432t^{24} + 10368t^{22} - 72576t^{20} + 96768t^{18} + 3877632t^{16} - 28864512t^{14} - 28643328t^{12} - 115458048t^{10} + 62042112t^{8} + 6193152t^{6} - 18579456t^{4} + 10616832t^{2} - 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 5377x - 149855$, with conductor $1344$
Generic density of odd order reductions $271/2688$