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

Curve name $X_{185k}$
Index $96$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 6 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$
Meaning/Special name
Chosen covering $X_{185}$
Curves that $X_{185k}$ minimally covers
Curves that minimally cover $X_{185k}$
Curves that minimally cover $X_{185k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{16} - 414720t^{12} - 231552t^{8} - 25920t^{4} - 108$ $B(t) = 1769472t^{24} - 55738368t^{20} - 115126272t^{16} - 48771072t^{12} - 7195392t^{8} - 217728t^{4} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 8641x - 311905$, with conductor $960$
Generic density of odd order reductions $299/2688$