| Curve name |
$X_{672}$ |
| 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} 29 & 0 \\ 2 & 3 \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 $X_{672}$ minimally covers |
$X_{308}$ |
| Curves that minimally cover $X_{672}$ |
|
| Curves that minimally cover $X_{672}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^3 + x^2 - 3x + 1\]\[w^2 = 1922x^2y^2 + 102132x^2y + 185728x^2 -
3064xy^2 - 195780xy - 244400x - 13361y^3 - 54210y^2 + 96784y + 58672\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-1/680 : -7/1530 : -1/6120 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 1/281 : -1/281 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(5/4 : -7/8 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : 0 : 0 : 1)$ |
Singular
|
| $(1 : 0 : 1 : 0)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(1/680 : 7/1530 : 1/6120 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(0 : -1/281 : 1/281 : 1)$ |
\[1728 \,\,(\text{CM by }-4)\]
|
| $(2529/961 : 127288/29791 : 1 : 0)$ |
Singular
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{689}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
None |
| Generic density of odd order reductions |
N/A |