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

Curve name $X_{118d}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 16 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{118d}$ minimally covers
Curves that minimally cover $X_{118d}$
Curves that minimally cover $X_{118d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{14} + 216t^{12} - 3456t^{8} + 6480t^{6} + 3456t^{4} - 6912t^{2}$ $B(t) = 54t^{21} - 648t^{19} + 1296t^{17} + 12096t^{15} - 55728t^{13} + 5184t^{11} + 314496t^{9} - 456192t^{7} + 165888t^{5} - 221184t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 24755x - 1492878$, with conductor $2925$
Generic density of odd order reductions $307/2688$