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

Curve name $X_{240o}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36a}$ $16$ $48$ $X_{118t}$
Meaning/Special name
Chosen covering $X_{240}$
Curves that $X_{240o}$ minimally covers
Curves that minimally cover $X_{240o}$
Curves that minimally cover $X_{240o}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{24} + 216t^{20} - 3456t^{12} + 6480t^{8} + 3456t^{4} - 6912$ $B(t) = 54t^{36} - 648t^{32} + 1296t^{28} + 12096t^{24} - 55728t^{20} + 5184t^{16} + 314496t^{12} - 456192t^{8} + 165888t^{4} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 5$, with conductor $45$
Generic density of odd order reductions $299/2688$