## The modular curve $X_{84p}$

Curve name $X_{84p}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 5 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{84p}$ minimally covers
Curves that minimally cover $X_{84p}$
Curves that minimally cover $X_{84p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 216t^{10} - 270t^{8} + 1404t^{6} + 4077t^{4} + 2916t^{2} - 108$ $B(t) = -54t^{18} - 648t^{16} - 5022t^{14} - 26460t^{12} - 82458t^{10} - 138672t^{8} - 103194t^{6} + 324t^{4} + 29160t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 2892x - 512624$, with conductor $2880$
Generic density of odd order reductions $41/336$