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

Curve name $X_{85o}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{85o}$ minimally covers
Curves that minimally cover $X_{85o}$
Curves that minimally cover $X_{85o}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} + 1728t^{10} - 4320t^{8} - 44928t^{6} + 260928t^{4} - 373248t^{2} - 27648$ $B(t) = 432t^{18} - 10368t^{16} + 160704t^{14} - 1693440t^{12} + 10554624t^{10} - 35500032t^{8} + 52835328t^{6} + 331776t^{4} - 59719680t^{2} + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 19596x + 2960624$, with conductor $4032$
Generic density of odd order reductions $635/5376$