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

Curve name $X_{99o}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 4 & 3 \end{matrix}\right]$
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 $X_{99o}$ minimally covers
Curves that minimally cover $X_{99o}$
Curves that minimally cover $X_{99o}$ 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 $600$
Generic density of odd order reductions $41/336$