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

Curve name $X_{99c}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 4 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{99}$
Curves that $X_{99c}$ minimally covers
Curves that minimally cover $X_{99c}$
Curves that minimally cover $X_{99c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 1080t^{14} - 4536t^{12} - 10368t^{10} - 14040t^{8} - 11664t^{6} - 6156t^{4} - 2160t^{2} - 432$ $B(t) = 432t^{24} + 6480t^{22} + 43416t^{20} + 171288t^{18} + 440640t^{16} + 769824t^{14} + 914760t^{12} + 705672t^{10} + 296784t^{8} + 5616t^{6} - 59616t^{4} - 25920t^{2} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 45075x + 3662750$, with conductor $1800$
Generic density of odd order reductions $691/5376$