## The modular curve $X_{185h}$

Curve name $X_{185h}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 2 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25h}$
Meaning/Special name
Chosen covering $X_{185}$
Curves that $X_{185h}$ minimally covers
Curves that minimally cover $X_{185h}$
Curves that minimally cover $X_{185h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -6912t^{16} - 103680t^{12} - 57888t^{8} - 6480t^{4} - 27$ $B(t) = 221184t^{24} - 6967296t^{20} - 14390784t^{16} - 6096384t^{12} - 899424t^{8} - 27216t^{4} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 135x - 660$, with conductor $15$
Generic density of odd order reductions $19/336$