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

Curve name $X_{84n}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{84n}$ minimally covers
Curves that minimally cover $X_{84n}$
Curves that minimally cover $X_{84n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 864t^{10} - 1080t^{8} + 5616t^{6} + 16308t^{4} + 11664t^{2} - 432$ $B(t) = -432t^{18} - 5184t^{16} - 40176t^{14} - 211680t^{12} - 659664t^{10} - 1109376t^{8} - 825552t^{6} + 2592t^{4} + 233280t^{2} + 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 723x - 64078$, with conductor $720$
Generic density of odd order reductions $635/5376$