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

Curve name $X_{185c}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
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 $X_{185c}$ minimally covers
Curves that minimally cover $X_{185c}$
Curves that minimally cover $X_{185c}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -7077888t^{32} - 106168320t^{28} - 58392576t^{24} + 6635520t^{20} + 7354368t^{16} + 414720t^{12} - 228096t^{8} - 25920t^{4} - 108$ $B(t) = 7247757312t^{48} - 228304355328t^{44} - 472916164608t^{40} - 156959244288t^{36} + 59029585920t^{32} + 33888927744t^{28} - 2118057984t^{20} - 230584320t^{16} + 38320128t^{12} + 7216128t^{8} + 217728t^{4} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 1944300x + 1042958000$, with conductor $14400$
Generic density of odd order reductions $51/448$