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

Curve name $X_{100g}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 4 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 4 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$
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 $X_{100g}$ minimally covers
Curves that minimally cover $X_{100g}$
Curves that minimally cover $X_{100g}$ 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 $12936$
Generic density of odd order reductions $691/5376$