## The modular curve $X_{99p}$

Curve name $X_{99p}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25i}$
Meaning/Special name
Chosen covering $X_{99}$
Curves that $X_{99p}$ minimally covers
Curves that minimally cover $X_{99p}$
Curves that minimally cover $X_{99p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 216t^{10} - 675t^{8} - 1026t^{6} - 783t^{4} - 324t^{2} - 108$ $B(t) = 54t^{18} + 648t^{16} + 3321t^{14} + 9450t^{12} + 16119t^{10} + 16200t^{8} + 7938t^{6} - 324t^{4} - 1944t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 7212x + 234416$, with conductor $2880$
Generic density of odd order reductions $25/224$