## The modular curve $X_{192i}$

Curve name $X_{192i}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25h}$
Meaning/Special name
Chosen covering $X_{192}$
Curves that $X_{192i}$ minimally covers
Curves that minimally cover $X_{192i}$
Curves that minimally cover $X_{192i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 25056t^{14} - 316224t^{12} + 2059776t^{10} - 4907520t^{8} + 32956416t^{6} - 80953344t^{4} - 102629376t^{2} - 1769472$ $B(t) = 54t^{24} - 111456t^{22} - 9979200t^{20} - 8805888t^{18} + 75852288t^{16} + 3849928704t^{14} - 25856409600t^{12} + 61598859264t^{10} + 19418185728t^{8} - 36068917248t^{6} - 653996851200t^{4} - 116870086656t^{2} + 905969664$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 120050x - 16020000$, with conductor $210$
Generic density of odd order reductions $1/28$