## The modular curve $X_{120k}$

Curve name $X_{120k}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36o}$
Meaning/Special name
Chosen covering $X_{120}$
Curves that $X_{120k}$ minimally covers
Curves that minimally cover $X_{120k}$
Curves that minimally cover $X_{120k}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{12} + 2592t^{10} - 24192t^{8} + 110592t^{6} - 250560t^{4} + 235008t^{2} - 27648$ $B(t) = -432t^{18} + 15552t^{16} - 238464t^{14} + 2032128t^{12} - 10502784t^{10} + 33550848t^{8} - 64060416t^{6} + 65359872t^{4} - 25214976t^{2} - 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 993600x - 380962764$, with conductor $2070$
Generic density of odd order reductions $193/1792$