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

Curve name $X_{84d}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
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 $X_{84d}$ minimally covers
Curves that minimally cover $X_{84d}$
Curves that minimally cover $X_{84d}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 162t^{10} + 27t^{8} + 1188t^{6} + 1755t^{4} + 702t^{2} - 27$ $B(t) = -54t^{18} - 486t^{16} - 3564t^{14} - 15876t^{12} - 35640t^{10} - 37584t^{8} - 12852t^{6} + 5508t^{4} + 3726t^{2} + 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 2008x - 295988$, with conductor $600$
Generic density of odd order reductions $41/336$