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

Curve name $X_{188d}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 2 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25h}$
Meaning/Special name
Chosen covering $X_{188}$
Curves that $X_{188d}$ minimally covers
Curves that minimally cover $X_{188d}$
Curves that minimally cover $X_{188d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} + 432t^{14} - 1296t^{12} - 1728t^{10} - 99360t^{8} - 6912t^{6} - 20736t^{4} + 27648t^{2} - 6912$ $B(t) = -54t^{24} + 1296t^{22} - 9072t^{20} + 12096t^{18} + 484704t^{16} - 3608064t^{14} - 3580416t^{12} - 14432256t^{10} + 7755264t^{8} + 774144t^{6} - 2322432t^{4} + 1327104t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 1344x - 19404$, with conductor $336$
Generic density of odd order reductions $53/896$