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

Curve name $X_{84a}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$
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 $X_{84a}$ minimally covers
Curves that minimally cover $X_{84a}$
Curves that minimally cover $X_{84a}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} - 648t^{10} + 108t^{8} + 4752t^{6} + 7020t^{4} + 2808t^{2} - 108$ $B(t) = -432t^{18} - 3888t^{16} - 28512t^{14} - 127008t^{12} - 285120t^{10} - 300672t^{8} - 102816t^{6} + 44064t^{4} + 29808t^{2} + 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 8033x - 2375937$, with conductor $4800$
Generic density of odd order reductions $635/5376$