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

Curve name $X_{240e}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 0 \\ 16 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 16 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36c}$ $16$ $48$ $X_{118a}$
Meaning/Special name
Chosen covering $X_{240}$
Curves that $X_{240e}$ minimally covers
Curves that minimally cover $X_{240e}$
Curves that minimally cover $X_{240e}$ 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 $y^2 = x^3 - 1152300x - 476098000$, with conductor $14400$
Generic density of odd order reductions $51/448$