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

Curve name $X_{120n}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right]$
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 $X_{120n}$ minimally covers
Curves that minimally cover $X_{120n}$
Curves that minimally cover $X_{120n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} + 4320t^{14} - 72576t^{12} + 663552t^{10} - 3568320t^{8} + 11321856t^{6} - 19823616t^{4} + 15482880t^{2} - 1769472$ $B(t) = -432t^{24} + 25920t^{22} - 694656t^{20} + 10962432t^{18} - 113021568t^{16} + 797879808t^{14} - 3926264832t^{12} + 13421998080t^{10} - 31071485952t^{8} + 45951418368t^{6} - 38263062528t^{4} + 12570329088t^{2} + 905969664$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - x^2 - 185220230x - 970197193353$, with conductor $22050$
Generic density of odd order reductions $193/1792$