## The modular curve $X_{121}$

Curve name $X_{121}$
Index $24$
Level $16$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right], \left[ \begin{matrix} 3 & 0 \\ 8 & 3 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$
Images in lower levels
 Level Index of image Corresponding curve $2$ $3$ $X_{6}$ $4$ $6$ $X_{13}$ $8$ $12$ $X_{36}$
Meaning/Special name
Chosen covering $X_{36}$
Curves that $X_{121}$ minimally covers $X_{36}$
Curves that minimally cover $X_{121}$ $X_{208}$, $X_{223}$, $X_{225}$, $X_{234}$, $X_{339}$, $X_{342}$, $X_{121a}$, $X_{121b}$, $X_{121c}$, $X_{121d}$, $X_{121e}$, $X_{121f}$, $X_{121g}$, $X_{121h}$, $X_{121i}$, $X_{121j}$, $X_{121k}$, $X_{121l}$, $X_{121m}$, $X_{121n}$, $X_{121o}$, $X_{121p}$
Curves that minimally cover $X_{121}$ and have infinitely many rational points. $X_{208}$, $X_{223}$, $X_{225}$, $X_{234}$, $X_{121a}$, $X_{121b}$, $X_{121c}$, $X_{121d}$, $X_{121e}$, $X_{121f}$, $X_{121g}$, $X_{121h}$, $X_{121i}$, $X_{121j}$, $X_{121k}$, $X_{121l}$, $X_{121m}$, $X_{121n}$, $X_{121o}$, $X_{121p}$
Model $\mathbb{P}^{1}, \mathbb{Q}(X_{121}) = \mathbb{Q}(f_{121}), f_{36} = -2f_{121}^{2} - 4$
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy = x^3 - x^2 - 181566x + 29819155$, with conductor $1989$
Generic density of odd order reductions $83/672$