| Curve name |
$X_{27}$ |
| Index |
$12$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$3$ |
$X_{6}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{10}$ |
| Curves that $X_{27}$ minimally covers |
$X_{10}$, $X_{11}$, $X_{13}$ |
| Curves that minimally cover $X_{27}$ |
$X_{60}$, $X_{92}$, $X_{94}$, $X_{95}$, $X_{134}$, $X_{135}$, $X_{27a}$, $X_{27b}$, $X_{27c}$, $X_{27d}$, $X_{27e}$, $X_{27f}$, $X_{27g}$, $X_{27h}$ |
| Curves that minimally cover $X_{27}$ and have infinitely many rational
points. |
$X_{60}$, $X_{92}$, $X_{94}$, $X_{95}$, $X_{27a}$, $X_{27b}$, $X_{27c}$, $X_{27d}$, $X_{27e}$, $X_{27f}$, $X_{27g}$, $X_{27h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{27}) = \mathbb{Q}(f_{27}), f_{10} =
\frac{f_{27}}{f_{27}^{2} - \frac{1}{4}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 885x - 4294$, with conductor $1845$ |
| Generic density of odd order reductions |
$9/56$ |