## The modular curve $X_{117g}$

Curve name $X_{117g}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36q}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117g}$ minimally covers
Curves that minimally cover $X_{117g}$
Curves that minimally cover $X_{117g}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -432t^{12} + 432t^{10} + 1620t^{8} - 1728t^{6} + 432t^{2} - 108$ $B(t) = -3456t^{18} + 5184t^{16} - 28512t^{14} + 39312t^{12} + 1296t^{10} - 27864t^{8} + 12096t^{6} + 2592t^{4} - 2592t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 26672x - 30440388$, with conductor $12936$
Generic density of odd order reductions $691/5376$