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

Curve name $X_{120f}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13h}$ $8$ $24$ $X_{36h}$
Meaning/Special name
Chosen covering $X_{120}$
Curves that $X_{120f}$ minimally covers
Curves that minimally cover $X_{120f}$
Curves that minimally cover $X_{120f}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} + 864t^{10} - 10800t^{8} + 65664t^{6} - 193968t^{4} + 228096t^{2} - 27648$ $B(t) = 54t^{18} - 2592t^{16} + 53136t^{14} - 604800t^{12} + 4153680t^{10} - 17459712t^{8} + 43182720t^{6} - 55655424t^{4} + 25878528t^{2} + 1769472$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 + x^2 + 48x + 48$, with conductor $147$
Generic density of odd order reductions $25/224$