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

Curve name $X_{120m}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36a}$
Meaning/Special name
Chosen covering $X_{120}$
Curves that $X_{120m}$ minimally covers
Curves that minimally cover $X_{120m}$
Curves that minimally cover $X_{120m}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} + 1080t^{14} - 18144t^{12} + 165888t^{10} - 892080t^{8} + 2830464t^{6} - 4955904t^{4} + 3870720t^{2} - 442368$ $B(t) = 54t^{24} - 3240t^{22} + 86832t^{20} - 1370304t^{18} + 14127696t^{16} - 99734976t^{14} + 490783104t^{12} - 1677749760t^{10} + 3883935744t^{8} - 5743927296t^{6} + 4782882816t^{4} - 1571291136t^{2} - 113246208$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 + 432x - 869$, with conductor $441$
Generic density of odd order reductions $25/224$