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

Curve name $X_{121e}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 7 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 5 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 8 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $12$ $X_{13f}$ $8$ $24$ $X_{36f}$
Meaning/Special name
Chosen covering $X_{121}$
Curves that $X_{121e}$ minimally covers
Curves that minimally cover $X_{121e}$
Curves that minimally cover $X_{121e}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 432t^{10} - 2700t^{8} - 8208t^{6} - 12123t^{4} - 7128t^{2} - 432$ $B(t) = -54t^{18} - 1296t^{16} - 13284t^{14} - 75600t^{12} - 259605t^{10} - 545616t^{8} - 674730t^{6} - 434808t^{4} - 101088t^{2} + 3456$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 + x^2 - 383x - 3012$, with conductor $1200$
Generic density of odd order reductions $25/224$