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

Curve name $X_{84f}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 1 \end{matrix}\right]$
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 $X_{84f}$ minimally covers
Curves that minimally cover $X_{84f}$
Curves that minimally cover $X_{84f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 1080t^{14} - 2916t^{12} + 2592t^{10} + 26460t^{8} + 49896t^{6} + 39204t^{4} + 10800t^{2} - 432$ $B(t) = -432t^{24} - 6480t^{22} - 57024t^{20} - 348192t^{18} - 1420416t^{16} - 3763584t^{14} - 6344352t^{12} - 6461856t^{10} - 3344976t^{8} - 114480t^{6} + 712800t^{4} + 243648t^{2} + 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 18075x - 8009750$, with conductor $3600$
Generic density of odd order reductions $635/5376$