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

Curve name $X_{117b}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $24$ $X_{36h}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117b}$ minimally covers
Curves that minimally cover $X_{117b}$
Curves that minimally cover $X_{117b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -432t^{16} + 1944t^{12} - 1323t^{8} + 324t^{4} - 27$ $B(t) = -3456t^{24} - 23328t^{20} + 39528t^{16} - 23814t^{12} + 6885t^{8} - 972t^{4} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 12328367x + 16665315108$, with conductor $142296$
Generic density of odd order reductions $307/2688$