## The modular curve $X_{118w}$

Curve name $X_{118w}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36s}$
Meaning/Special name
Chosen covering $X_{118}$
Curves that $X_{118w}$ minimally covers
Curves that minimally cover $X_{118w}$
Curves that minimally cover $X_{118w}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 216t^{10} + 3456t^{6} + 6480t^{4} - 3456t^{2} - 6912$ $B(t) = -54t^{18} - 648t^{16} - 1296t^{14} + 12096t^{12} + 55728t^{10} + 5184t^{8} - 314496t^{6} - 456192t^{4} - 165888t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 297496x - 62354384$, with conductor $40560$
Generic density of odd order reductions $41/336$