| Curve name |
$X_{30}$ |
| Index |
$12$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{12}$ |
| Curves that $X_{30}$ minimally covers |
$X_{12}$ |
| Curves that minimally cover $X_{30}$ |
$X_{72}$, $X_{88}$, $X_{89}$, $X_{93}$, $X_{103}$, $X_{104}$, $X_{126}$, $X_{142}$, $X_{162}$, $X_{163}$, $X_{164}$, $X_{167}$, $X_{172}$, $X_{174}$ |
| Curves that minimally cover $X_{30}$ and have infinitely many rational
points. |
$X_{89}$, $X_{93}$, $X_{103}$, $X_{104}$, $X_{167}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{30}) = \mathbb{Q}(f_{30}), f_{12} =
-f_{30}^{2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 7325x + 219000$, with conductor $63075$ |
| Generic density of odd order reductions |
$419/1344$ |