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

Curve name $X_{185a}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
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 $X_{185a}$ minimally covers
Curves that minimally cover $X_{185a}$
Curves that minimally cover $X_{185a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -442368t^{24} - 6414336t^{20} - 414720t^{16} + 1022976t^{12} - 25920t^{8} - 25056t^{4} - 108$ $B(t) = 113246208t^{36} - 3652190208t^{32} - 4671406080t^{28} + 1734082560t^{24} + 554729472t^{20} - 138682368t^{16} - 27095040t^{12} + 4561920t^{8} + 222912t^{4} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 77772x + 8343664$, with conductor $2880$
Generic density of odd order reductions $73/672$