| Curve name |
$X_{123}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 11 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 13 & 13 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 6 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{37}$ |
| Curves that $X_{123}$ minimally covers |
$X_{37}$ |
| Curves that minimally cover $X_{123}$ |
$X_{239}$, $X_{123a}$, $X_{123b}$, $X_{123c}$, $X_{123d}$, $X_{123e}$, $X_{123f}$ |
| Curves that minimally cover $X_{123}$ and have infinitely many rational
points. |
$X_{239}$, $X_{123a}$, $X_{123b}$, $X_{123c}$, $X_{123d}$, $X_{123e}$, $X_{123f}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{123}) = \mathbb{Q}(f_{123}), f_{37} =
\frac{f_{123} + 1}{f_{123}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 6281x - 186919$, with conductor $10880$ |
| Generic density of odd order reductions |
$85091/344064$ |