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

Curve name $X_{118v}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$ $16$ $24$ $X_{118}$
Meaning/Special name
Chosen covering $X_{118}$
Curves that $X_{118v}$ minimally covers
Curves that minimally cover $X_{118v}$
Curves that minimally cover $X_{118v}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{10} + 1728t^{6} - 1728t^{2}$ $B(t) = 432t^{15} - 10368t^{11} + 51840t^{7} + 27648t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 + x^2 - 3250000x - 2256492875$, with conductor $975$
Generic density of odd order reductions $25/224$