| Curve name |
$X_{231}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 12 & 15 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 12 & 15 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{94}$ |
| Curves that $X_{231}$ minimally covers |
$X_{94}$, $X_{110}$, $X_{112}$ |
| Curves that minimally cover $X_{231}$ |
$X_{231a}$, $X_{231b}$, $X_{231c}$, $X_{231d}$ |
| Curves that minimally cover $X_{231}$ and have infinitely many rational
points. |
$X_{231a}$, $X_{231b}$, $X_{231c}$, $X_{231d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{231}) = \mathbb{Q}(f_{231}), f_{94} =
\frac{8f_{231} + 8}{f_{231}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 24009700x + 65021152000$, with conductor $421600$ |
| Generic density of odd order reductions |
$9249/57344$ |