## The modular curve $X_{85k}$

Curve name $X_{85k}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{85k}$ minimally covers
Curves that minimally cover $X_{85k}$
Curves that minimally cover $X_{85k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{8} + 216t^{6} + 1080t^{4} - 6048t^{2} - 432$ $B(t) = 54t^{12} - 648t^{10} + 9720t^{8} - 60480t^{6} + 85536t^{4} + 114048t^{2} - 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 370x + 2435$, with conductor $210$
Generic density of odd order reductions $47/672$