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

Curve name $X_{84j}$
Index $48$
Level $8$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right]$
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 $X_{84j}$ minimally covers
Curves that minimally cover $X_{84j}$
Curves that minimally cover $X_{84j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{8} - 432t^{6} + 1080t^{4} + 3024t^{2} - 108$ $B(t) = 432t^{12} + 2592t^{10} + 19440t^{8} + 60480t^{6} + 42768t^{4} - 28512t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 321x + 18879$, with conductor $960$
Generic density of odd order reductions $635/5376$