| Curve name |
$X_{122}$ |
| 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 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{36}$ |
| Curves that $X_{122}$ minimally covers |
$X_{36}$ |
| Curves that minimally cover $X_{122}$ |
$X_{207}$, $X_{208}$, $X_{211}$, $X_{212}$, $X_{217}$, $X_{228}$, $X_{230}$, $X_{233}$, $X_{335}$, $X_{337}$, $X_{346}$, $X_{347}$, $X_{122a}$, $X_{122b}$, $X_{122c}$, $X_{122d}$, $X_{122e}$, $X_{122f}$, $X_{122g}$, $X_{122h}$, $X_{122i}$, $X_{122j}$, $X_{122k}$, $X_{122l}$, $X_{122m}$, $X_{122n}$, $X_{122o}$, $X_{122p}$ |
| Curves that minimally cover $X_{122}$ and have infinitely many rational
points. |
$X_{207}$, $X_{208}$, $X_{211}$, $X_{212}$, $X_{217}$, $X_{228}$, $X_{230}$, $X_{233}$, $X_{122a}$, $X_{122b}$, $X_{122c}$, $X_{122d}$, $X_{122e}$, $X_{122f}$, $X_{122g}$, $X_{122h}$, $X_{122i}$, $X_{122j}$, $X_{122k}$, $X_{122l}$, $X_{122m}$, $X_{122n}$, $X_{122o}$, $X_{122p}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{122}) = \mathbb{Q}(f_{122}), f_{36} =
8f_{122}^{2} - 4\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 19600x - 1064375$, with conductor $525$ |
| Generic density of odd order reductions |
$19/168$ |