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

Curve name $X_{85h}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{85}$
Meaning/Special name
Chosen covering $X_{85}$
Curves that $X_{85h}$ minimally covers
Curves that minimally cover $X_{85h}$
Curves that minimally cover $X_{85h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} + 2160t^{14} - 11664t^{12} - 20736t^{10} + 423360t^{8} - 1596672t^{6} + 2509056t^{4} - 1382400t^{2} - 110592$ $B(t) = -432t^{24} + 12960t^{22} - 228096t^{20} + 2785536t^{18} - 22726656t^{16} + 120434688t^{14} - 406038528t^{12} + 827117568t^{10} - 856313856t^{8} + 58613760t^{6} + 729907200t^{4} - 498991104t^{2} + 14155776$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 15003x - 1979636$, with conductor $441$
Generic density of odd order reductions $25/224$