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

Curve name $X_{85j}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
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 $X_{85j}$ minimally covers
Curves that minimally cover $X_{85j}$
Curves that minimally cover $X_{85j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} + 864t^{6} + 4320t^{4} - 24192t^{2} - 1728$ $B(t) = -432t^{12} + 5184t^{10} - 77760t^{8} + 483840t^{6} - 684288t^{4} - 912384t^{2} + 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 2177x - 108927$, with conductor $1344$
Generic density of odd order reductions $635/5376$