## The modular curve $X_{185}$

Curve name $X_{185}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{87}$
Curves that $X_{185}$ minimally covers $X_{87}$, $X_{96}$, $X_{101}$
Curves that minimally cover $X_{185}$ $X_{442}$, $X_{452}$, $X_{456}$, $X_{458}$, $X_{485}$, $X_{486}$, $X_{185a}$, $X_{185b}$, $X_{185c}$, $X_{185d}$, $X_{185e}$, $X_{185f}$, $X_{185g}$, $X_{185h}$, $X_{185i}$, $X_{185j}$, $X_{185k}$, $X_{185l}$
Curves that minimally cover $X_{185}$ and have infinitely many rational points. $X_{185a}$, $X_{185b}$, $X_{185c}$, $X_{185d}$, $X_{185e}$, $X_{185f}$, $X_{185g}$, $X_{185h}$, $X_{185i}$, $X_{185j}$, $X_{185k}$, $X_{185l}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{185}) = \mathbb{Q}(f_{185}), f_{87} = \frac{f_{185}}{f_{185}^{2} - \frac{1}{2}}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - 6616x + 206471$, with conductor $735$
Generic density of odd order reductions $25/224$