## The modular curve $X_{240j}$

Curve name $X_{240j}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $24$ $X_{36h}$ $16$ $48$ $X_{118f}$
Meaning/Special name
Chosen covering $X_{240}$
Curves that $X_{240j}$ minimally covers
Curves that minimally cover $X_{240j}$
Curves that minimally cover $X_{240j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{32} + 1296t^{24} - 21168t^{16} + 124416t^{8} - 110592$ $B(t) = 54t^{48} - 3888t^{40} + 110160t^{32} - 1524096t^{24} + 10119168t^{16} - 23887872t^{8} - 14155776$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 5x - 628$, with conductor $225$
Generic density of odd order reductions $299/2688$