## The modular curve $X_{122n}$

Curve name $X_{122n}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36q}$
Meaning/Special name
Chosen covering $X_{122}$
Curves that $X_{122n}$ minimally covers
Curves that minimally cover $X_{122n}$
Curves that minimally cover $X_{122n}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27648t^{16} + 138240t^{14} - 290304t^{12} + 331776t^{10} - 223020t^{8} + 88452t^{6} - 19359t^{4} + 1890t^{2} - 27$ $B(t) = 1769472t^{24} - 13271040t^{22} + 44457984t^{20} - 87699456t^{18} + 113021568t^{16} - 99734976t^{14} + 61347888t^{12} - 26214840t^{10} + 7585812t^{8} - 1402326t^{6} + 145962t^{4} - 5994t^{2} - 54$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 5532051x + 5008150546$, with conductor $7056$
Generic density of odd order reductions $41/336$