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

Curve name $X_{117k}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $24$ $X_{36l}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117k}$ minimally covers
Curves that minimally cover $X_{117k}$
Curves that minimally cover $X_{117k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 108t^{4} - 27$ $B(t) = -54t^{12} - 405t^{8} + 324t^{4} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 2079x - 35802$, with conductor $3696$
Generic density of odd order reductions $65/896$