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

Curve name $X_{85m}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{85m}$ minimally covers
Curves that minimally cover $X_{85m}$
Curves that minimally cover $X_{85m}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} + 432t^{10} - 1080t^{8} - 11232t^{6} + 65232t^{4} - 93312t^{2} - 6912$ $B(t) = 54t^{18} - 1296t^{16} + 20088t^{14} - 211680t^{12} + 1319328t^{10} - 4437504t^{8} + 6604416t^{6} + 41472t^{4} - 7464960t^{2} + 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 306x + 5859$, with conductor $63$
Generic density of odd order reductions $17/168$