| Curve name |
$X_{239}$ |
| Index |
$48$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 27 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 29 & 29 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 31 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{123}$ |
| Curves that $X_{239}$ minimally covers |
$X_{123}$ |
| Curves that minimally cover $X_{239}$ |
$X_{239a}$, $X_{239b}$, $X_{239c}$, $X_{239d}$ |
| Curves that minimally cover $X_{239}$ and have infinitely many rational
points. |
$X_{239a}$, $X_{239b}$, $X_{239c}$, $X_{239d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{239}) = \mathbb{Q}(f_{239}), f_{123} =
\frac{4f_{239} + 4}{f_{239}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 84x + 272$, with conductor $1152$ |
| Generic density of odd order reductions |
$2722915/11010048$ |