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

Curve name $X_{121m}$
Index $48$
Level $16$
Genus $0$
Does the subgroup contain $-I$? No
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 5 \\ 0 & 1 \end{matrix}\right]$
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 $X_{121m}$ minimally covers
Curves that minimally cover $X_{121m}$
Curves that minimally cover $X_{121m}$ and have infinitely many rational points.
Model $\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is given by $y^2 = x^3 + A(t)x + B(t), \text{ where}$ $A(t) = -27t^{16} - 540t^{14} - 4536t^{12} - 20736t^{10} - 55755t^{8} - 88452t^{6} - 77436t^{4} - 30240t^{2} - 1728$ $B(t) = 54t^{24} + 1620t^{22} + 21708t^{20} + 171288t^{18} + 882981t^{16} + 3116718t^{14} + 7668486t^{12} + 13107420t^{10} + 15171624t^{8} + 11218608t^{6} + 4670784t^{4} + 767232t^{2} - 27648$
Elliptic curve whose $2$-adic image is the subgroup $y^2 = x^3 - 3450x + 77875$, with conductor $3600$
Generic density of odd order reductions $635/5376$