## The modular curve $X_{100f}$

Curve name $X_{100f}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{100}$
Curves that $X_{100f}$ minimally covers
Curves that minimally cover $X_{100f}$
Curves that minimally cover $X_{100f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} + 108t^{10} - 432t^{6} + 1728t^{2} - 1728$ $B(t) = 54t^{18} - 324t^{16} + 324t^{14} + 1512t^{12} - 5184t^{10} + 10368t^{8} - 12096t^{6} - 10368t^{4} + 41472t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 102132x + 12420540$, with conductor $25872$
Generic density of odd order reductions $193/1792$