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

Curve name $X_{99h}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25h}$
Meaning/Special name
Chosen covering $X_{99}$
Curves that $X_{99h}$ minimally covers
Curves that minimally cover $X_{99h}$
Curves that minimally cover $X_{99h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} - 108t^{6} - 135t^{4} - 54t^{2} - 27$ $B(t) = -54t^{12} - 324t^{10} - 729t^{8} - 756t^{6} - 243t^{4} + 162t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 200x - 1152$, with conductor $120$
Generic density of odd order reductions $47/672$