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

Curve name $X_{240p}$
Index $96$
Level $32$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 16 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36q}$ $16$ $48$ $X_{118x}$
Meaning/Special name
Chosen covering $X_{240}$
Curves that $X_{240p}$ minimally covers
Curves that minimally cover $X_{240p}$
Curves that minimally cover $X_{240p}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{24} - 864t^{20} + 13824t^{12} + 25920t^{8} - 13824t^{4} - 27648$ $B(t) = 432t^{36} + 5184t^{32} + 10368t^{28} - 96768t^{24} - 445824t^{20} - 41472t^{16} + 2515968t^{12} + 3649536t^{8} + 1327104t^{4} + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - x^2 - 33x + 11937$, with conductor $4800$
Generic density of odd order reductions $109/896$