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

Curve name $X_{117j}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36c}$
Meaning/Special name
Chosen covering $X_{117}$
Curves that $X_{117j}$ minimally covers
Curves that minimally cover $X_{117j}$
Curves that minimally cover $X_{117j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1728t^{16} + 7776t^{12} - 5292t^{8} + 1296t^{4} - 108$ $B(t) = 27648t^{24} + 186624t^{20} - 316224t^{16} + 190512t^{12} - 55080t^{8} + 7776t^{4} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 12909409x + 324219616895$, with conductor $1138368$
Generic density of odd order reductions $335/2688$