| Curve name |
$X_{20b}$ |
| Index |
$16$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 3 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 1 & 0 \end{matrix}\right]$ |
| Images in lower levels |
| Level | Index of image | Corresponding curve |
| $2$ |
$1$ |
$X_{1}$ |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{20}$ |
| Curves that $X_{20b}$ minimally covers |
|
| Curves that minimally cover $X_{20b}$ |
|
| Curves that minimally cover $X_{20b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{8} + 648t^{7} - 4212t^{6} - 2376t^{5} + 60102t^{4} + 79704t^{3} -
105732t^{2} - 235224t - 107811\]
\[B(t) = 54t^{12} - 1944t^{11} + 24300t^{10} - 97848t^{9} - 251262t^{8} +
1722384t^{7} + 4821768t^{6} - 8697456t^{5} - 64323558t^{4} - 140447736t^{3} -
157012020t^{2} - 90561240t - 21346578\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 + 4x - 1$, with conductor $162$ |
| Generic density of odd order reductions |
$71/168$ |