| Curve name |
$X_{649}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$3$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 29 & 0 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 31 & 27 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 4 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 31 & 31 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{323}$ |
| Curves that $X_{649}$ minimally covers |
$X_{323}$ |
| Curves that minimally cover $X_{649}$ |
|
| Curves that minimally cover $X_{649}$ and have infinitely many rational
points. |
|
| Model |
\[x^4 - x^2y^2 - 2x^2z^2 - y^3z + 2yz^3 = 0\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(-2 : -4 : 1)$ |
\[\frac{777228872334890625}{60523872256}\]
|
| $(0 : 1 : 0)$ |
\[ \infty \]
|
| $(0 : 0 : 1)$ |
\[ \infty \]
|
| $(1 : 1 : 0)$ |
\[-3375 \,\,(\text{CM by }-7)\]
|
| $(2 : -4 : 1)$ |
\[\frac{777228872334890625}{60523872256}\]
|
| $(-1 : 1 : 0)$ |
\[-3375 \,\,(\text{CM by }-7)\]
|
|
| Comments on finding rational points |
This curve is isomorphic to $X_{619}$. |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 - x^2 - 325630x - 71434867$, with conductor $17918$ |
| Generic density of odd order reductions |
$410657/1835008$ |