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

Curve name $X_{240i}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 0 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 1 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36e}$ $16$ $48$ $X_{118e}$
Meaning/Special name
Chosen covering $X_{240}$
Curves that $X_{240i}$ minimally covers
Curves that minimally cover $X_{240i}$
Curves that minimally cover $X_{240i}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{32} + 5184t^{24} - 84672t^{16} + 497664t^{8} - 442368$ $B(t) = 432t^{48} - 31104t^{40} + 881280t^{32} - 12192768t^{24} + 80953344t^{16} - 191102976t^{8} - 113246208$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 300x - 322000$, with conductor $14400$
Generic density of odd order reductions $51/448$