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

Curve name $X_{84h}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{84}$
Meaning/Special name
Chosen covering $X_{84}$
Curves that $X_{84h}$ minimally covers
Curves that minimally cover $X_{84h}$
Curves that minimally cover $X_{84h}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 270t^{14} - 729t^{12} + 648t^{10} + 6615t^{8} + 12474t^{6} + 9801t^{4} + 2700t^{2} - 108$ $B(t) = 54t^{24} + 810t^{22} + 7128t^{20} + 43524t^{18} + 177552t^{16} + 470448t^{14} + 793044t^{12} + 807732t^{10} + 418122t^{8} + 14310t^{6} - 89100t^{4} - 30456t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 72300x + 64078000$, with conductor $14400$
Generic density of odd order reductions $41/336$