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

Curve name $X_{118o}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36o}$
Meaning/Special name
Chosen covering $X_{118}$
Curves that $X_{118o}$ minimally covers
Curves that minimally cover $X_{118o}$
Curves that minimally cover $X_{118o}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} + 864t^{10} - 13824t^{6} + 25920t^{4} + 13824t^{2} - 27648$ $B(t) = -432t^{18} + 5184t^{16} - 10368t^{14} - 96768t^{12} + 445824t^{10} - 41472t^{8} - 2515968t^{6} + 3649536t^{4} - 1327104t^{2} + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 71370x + 8011575$, with conductor $585$
Generic density of odd order reductions $25/224$