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

Curve name $X_{121j}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $24$ $X_{36a}$
Meaning/Special name
Chosen covering $X_{121}$
Curves that $X_{121j}$ minimally covers
Curves that minimally cover $X_{121j}$
Curves that minimally cover $X_{121j}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{12} - 324t^{10} - 1512t^{8} - 3456t^{6} - 3915t^{4} - 1836t^{2} - 108$ $B(t) = 54t^{18} + 972t^{16} + 7452t^{14} + 31752t^{12} + 82053t^{10} + 131058t^{8} + 125118t^{6} + 63828t^{4} + 12312t^{2} - 432$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 138x + 623$, with conductor $720$
Generic density of odd order reductions $635/5376$