## The modular curve $X_{185b}$

Curve name $X_{185b}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $48$ $X_{185}$
Meaning/Special name
Chosen covering $X_{185}$
Curves that $X_{185b}$ minimally covers
Curves that minimally cover $X_{185b}$
Curves that minimally cover $X_{185b}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -1769472t^{32} - 26542080t^{28} - 14598144t^{24} + 1658880t^{20} + 1838592t^{16} + 103680t^{12} - 57024t^{8} - 6480t^{4} - 27$ $B(t) = 905969664t^{48} - 28538044416t^{44} - 59114520576t^{40} - 19619905536t^{36} + 7378698240t^{32} + 4236115968t^{28} - 264757248t^{20} - 28823040t^{16} + 4790016t^{12} + 902016t^{8} + 27216t^{4} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 30380x + 2044622$, with conductor $225$
Generic density of odd order reductions $299/2688$