| Curve name |
$X_{20}$ |
| Index |
$8$ |
| 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} 1 & 1 \\ 3 & 0 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
|
| Meaning/Special name |
Elliptic curves whose discriminant is minus a square with $a_{p}(E) \equiv 0
\pmod{4}$ for $p \equiv 3 \pmod{4}$ |
| Chosen covering |
$X_{7}$ |
| Curves that $X_{20}$ minimally covers |
$X_{3}$, $X_{7}$ |
| Curves that minimally cover $X_{20}$ |
$X_{60}$, $X_{180}$, $X_{20a}$, $X_{20b}$ |
| Curves that minimally cover $X_{20}$ and have infinitely many rational
points. |
$X_{60}$, $X_{20a}$, $X_{20b}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{20}) = \mathbb{Q}(f_{20}), f_{7} =
\frac{f_{20} + 1}{f_{20}^{2} - 3}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 6x + 8$, with conductor $162$ |
| Generic density of odd order reductions |
$37/84$ |