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

Curve name $X_{85d}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right]$
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 $X_{85d}$ minimally covers
Curves that minimally cover $X_{85d}$
Curves that minimally cover $X_{85d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} + 1296t^{10} + 432t^{8} - 38016t^{6} + 112320t^{4} - 89856t^{2} - 6912$ $B(t) = 432t^{18} - 7776t^{16} + 114048t^{14} - 1016064t^{12} + 4561920t^{10} - 9621504t^{8} + 6580224t^{6} + 5640192t^{4} - 7630848t^{2} + 221184$
Info about rational points
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 - 1667x + 72764$, with conductor $147$
Generic density of odd order reductions $25/224$