| Curve name |
$X_{10}$ |
| Index |
$6$ |
| Level |
$4$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 1 \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 |
Elliptic curves that acquire full $2$-torsion over $\mathbb{Q}(i)$ |
| Chosen covering |
$X_{6}$ |
| Curves that $X_{10}$ minimally covers |
$X_{3}$, $X_{6}$ |
| Curves that minimally cover $X_{10}$ |
$X_{27}$, $X_{42}$, $X_{10a}$, $X_{10b}$, $X_{10c}$, $X_{10d}$ |
| Curves that minimally cover $X_{10}$ and have infinitely many rational
points. |
$X_{27}$, $X_{42}$, $X_{10a}$, $X_{10b}$, $X_{10c}$, $X_{10d}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{10}) = \mathbb{Q}(f_{10}), f_{6} =
\frac{48f_{10}^{2} - 16}{f_{10}^{2} + 1}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 33x - 74$, with conductor $180$ |
| Generic density of odd order reductions |
$83/336$ |