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

Curve name $X_{117n}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36s}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117n}$ minimally covers
Curves that minimally cover $X_{117n}$
Curves that minimally cover $X_{117n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 108t^{10} + 405t^{8} + 432t^{6} - 108t^{2} - 27$ $B(t) = -432t^{18} - 648t^{16} - 3564t^{14} - 4914t^{12} + 162t^{10} + 3483t^{8} + 1512t^{6} - 324t^{4} - 324t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1164378x - 483603127$, with conductor $8280$
Generic density of odd order reductions $635/5376$