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

Curve name $X_{185d}$
Index $96$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right]$
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 $X_{185d}$ minimally covers
Curves that minimally cover $X_{185d}$
Curves that minimally cover $X_{185d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -110592t^{24} - 1603584t^{20} - 103680t^{16} + 255744t^{12} - 6480t^{8} - 6264t^{4} - 27$ $B(t) = 14155776t^{36} - 456523776t^{32} - 583925760t^{28} + 216760320t^{24} + 69341184t^{20} - 17335296t^{16} - 3386880t^{12} + 570240t^{8} + 27864t^{4} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 1215x + 16600$, with conductor $45$
Generic density of odd order reductions $299/2688$