| 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 |
|
| 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 |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{121}) = \mathbb{Q}(f_{121}), f_{36} =
-2f_{121}^{2} - 4\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| 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 |
$83/672$ |