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

Curve name $X_{185i}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $24$ $X_{25i}$
Meaning/Special name
Chosen covering $X_{185}$
Curves that $X_{185i}$ minimally covers
Curves that minimally cover $X_{185i}$
Curves that minimally cover $X_{185i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -110592t^{24} - 1714176t^{20} - 1762560t^{16} - 670464t^{12} - 110160t^{8} - 6696t^{4} - 27$ $B(t) = 14155776t^{36} - 435290112t^{32} - 1252786176t^{28} - 1164312576t^{24} - 529846272t^{20} - 132461568t^{16} - 18192384t^{12} - 1223424t^{8} - 26568t^{4} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 3376x - 75727$, with conductor $75$
Generic density of odd order reductions $11/112$