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

Curve name $X_{117f}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36a}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117f}$ minimally covers
Curves that minimally cover $X_{117f}$
Curves that minimally cover $X_{117f}$ 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 $16560$
Generic density of odd order reductions $635/5376$