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

Curve name $X_{118b}$
Index $48$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{118b}$ minimally covers
Curves that minimally cover $X_{118b}$
Curves that minimally cover $X_{118b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{14} + 864t^{12} - 13824t^{8} + 25920t^{6} + 13824t^{4} - 27648t^{2}$ $B(t) = 432t^{21} - 5184t^{19} + 10368t^{17} + 96768t^{15} - 445824t^{13} + 41472t^{11} + 2515968t^{9} - 3649536t^{7} + 1327104t^{5} - 1769472t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1584300x + 767522000$, with conductor $187200$
Generic density of odd order reductions $25/224$