## The modular curve $X_{121o}$

Curve name $X_{121o}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 3 & 3 \\ 0 & 7 \end{matrix}\right]$
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 $X_{121}$
Curves that $X_{121o}$ minimally covers
Curves that minimally cover $X_{121o}$
Curves that minimally cover $X_{121o}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -108t^{16} - 2160t^{14} - 18144t^{12} - 82944t^{10} - 223020t^{8} - 353808t^{6} - 309744t^{4} - 120960t^{2} - 6912$ $B(t) = 432t^{24} + 12960t^{22} + 173664t^{20} + 1370304t^{18} + 7063848t^{16} + 24933744t^{14} + 61347888t^{12} + 104859360t^{10} + 121372992t^{8} + 89748864t^{6} + 37366272t^{4} + 6137856t^{2} - 221184$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 13800x + 623000$, with conductor $14400$
Generic density of odd order reductions $89/672$