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

Curve name $X_{96p}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $24$ $X_{96}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{96p}$ minimally covers
Curves that minimally cover $X_{96p}$
Curves that minimally cover $X_{96p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{14} - 216t^{12} + 216t^{8} - 216t^{4} - 108t^{2}$ $B(t) = 432t^{21} + 1296t^{19} + 648t^{17} - 1512t^{15} - 2592t^{13} - 2592t^{11} - 1512t^{9} + 648t^{7} + 1296t^{5} + 432t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 174947x - 23777854$, with conductor $7605$
Generic density of odd order reductions $25/224$