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

Curve name $X_{120l}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36s}$
Meaning/Special name
Chosen covering $X_{120}$
Curves that $X_{120l}$ minimally covers
Curves that minimally cover $X_{120l}$
Curves that minimally cover $X_{120l}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} + 648t^{10} - 6048t^{8} + 27648t^{6} - 62640t^{4} + 58752t^{2} - 6912$ $B(t) = -54t^{18} + 1944t^{16} - 29808t^{14} + 254016t^{12} - 1312848t^{10} + 4193856t^{8} - 8007552t^{6} + 8169984t^{4} - 3151872t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + 141x - 142$, with conductor $1008$
Generic density of odd order reductions $25/224$