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

Curve name $X_{99n}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
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 $X_{99n}$ minimally covers
Curves that minimally cover $X_{99n}$
Curves that minimally cover $X_{99n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 162t^{10} - 378t^{8} - 432t^{6} - 270t^{4} - 108t^{2} - 27$ $B(t) = 54t^{18} + 486t^{16} + 1863t^{14} + 3969t^{12} + 5022t^{10} + 3564t^{8} + 945t^{6} - 405t^{4} - 324t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 5008x + 133988$, with conductor $1200$
Generic density of odd order reductions $25/224$