| Curve name |
$X_{673}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$5$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 21 & 30 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 9 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 30 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 11 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{308}$ |
| Curves that $X_{673}$ minimally covers |
$X_{308}$ |
| Curves that minimally cover $X_{673}$ |
|
| Curves that minimally cover $X_{673}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 3x + 1\]\[w^2 = -24012x^2y - 92864x^2 + 10044xy^2 +
152380xy + 114512x - 961y^3 - 30512y^2 - 117616y - 9104\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(2529/961 : 127288/29791 : 1 : 0)$ |
Singular
|
| $(0 : 0 : 0 : 1)$ |
Singular
|
| $(9 : 28 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{686}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |