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

Curve name $X_{99f}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{99}$
Curves that $X_{99f}$ minimally covers
Curves that minimally cover $X_{99f}$
Curves that minimally cover $X_{99f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 270t^{14} - 1134t^{12} - 2592t^{10} - 3510t^{8} - 2916t^{6} - 1539t^{4} - 540t^{2} - 108$ $B(t) = -54t^{24} - 810t^{22} - 5427t^{20} - 21411t^{18} - 55080t^{16} - 96228t^{14} - 114345t^{12} - 88209t^{10} - 37098t^{8} - 702t^{6} + 7452t^{4} + 3240t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 180300x - 29302000$, with conductor $14400$
Generic density of odd order reductions $89/672$