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

Curve name $X_{99j}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 4 & 5 \end{matrix}\right]$
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 $X_{99j}$ minimally covers
Curves that minimally cover $X_{99j}$
Curves that minimally cover $X_{99j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 864t^{10} - 2700t^{8} - 4104t^{6} - 3132t^{4} - 1296t^{2} - 432$ $B(t) = -432t^{18} - 5184t^{16} - 26568t^{14} - 75600t^{12} - 128952t^{10} - 129600t^{8} - 63504t^{6} + 2592t^{4} + 15552t^{2} + 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1803x - 29302$, with conductor $720$
Generic density of odd order reductions $635/5376$