## The modular curve $X_{96j}$

Curve name $X_{96j}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 8 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $6$ $X_{8}$ $4$ $12$ $X_{25}$ $8$ $24$ $X_{96}$
Meaning/Special name
Chosen covering $X_{96}$
Curves that $X_{96j}$ minimally covers
Curves that minimally cover $X_{96j}$
Curves that minimally cover $X_{96j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{14} - 54t^{12} + 54t^{8} - 54t^{4} - 27t^{2}$ $B(t) = 54t^{21} + 162t^{19} + 81t^{17} - 189t^{15} - 324t^{13} - 324t^{11} - 189t^{9} + 81t^{7} + 162t^{5} + 54t^{3}$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 11196588x - 12196654288$, with conductor $486720$
Generic density of odd order reductions $307/2688$