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

Curve name $X_{99k}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{99}$
Curves that $X_{99k}$ minimally covers
Curves that minimally cover $X_{99k}$
Curves that minimally cover $X_{99k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 648t^{10} - 1512t^{8} - 1728t^{6} - 1080t^{4} - 432t^{2} - 108$ $B(t) = 432t^{18} + 3888t^{16} + 14904t^{14} + 31752t^{12} + 40176t^{10} + 28512t^{8} + 7560t^{6} - 3240t^{4} - 2592t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 20033x + 1091937$, with conductor $4800$
Generic density of odd order reductions $691/5376$