## The modular curve $X_{96n}$

Curve name $X_{96n}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{96n}$ minimally covers
Curves that minimally cover $X_{96n}$
Curves that minimally cover $X_{96n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 216t^{10} + 216t^{6} - 216t^{2} - 108$ $B(t) = -432t^{18} - 1296t^{16} - 648t^{14} + 1512t^{12} + 2592t^{10} + 2592t^{8} + 1512t^{6} - 648t^{4} - 1296t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 1373129x - 619428769$, with conductor $2535$
Generic density of odd order reductions $17/168$