| Curve name |
$X_{441}$ |
| Index |
$64$ |
| Level |
$16$ |
| Genus |
$2$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 4 & 7 \\ 15 & 12 \end{matrix}\right],
\left[ \begin{matrix} 7 & 14 \\ 7 & 9 \end{matrix}\right],
\left[ \begin{matrix} 2 & 1 \\ 11 & 9 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
$X_{ns}^{+}(16)$ |
| Chosen covering |
$X_{55}$ |
| Curves that $X_{441}$ minimally covers |
$X_{55}$ |
| Curves that minimally cover $X_{441}$ |
|
| Curves that minimally cover $X_{441}$ and have infinitely many rational
points. |
|
| Model |
\[y^2 = x^6 + x^4 - 3x^2 + 1\] |
| Info about rational points |
| Rational point | Image on the $j$-line |
| $(1 : -1 : 0)$ |
\[-147197952000 \,\,(\text{CM by }-67)\]
|
| $(1 : 1 : 0)$ |
\[-32768 \,\,(\text{CM by }-11)\]
|
| $(-1 : 0 : 1)$ |
\[0 \,\,(\text{CM by }-3)\]
|
| $(0 : -1 : 1)$ |
\[-12288000 \,\,(\text{CM by }-27)\]
|
| $(0 : 1 : 1)$ |
\[-884736 \,\,(\text{CM by }-19)\]
|
| $(1 : 0 : 1)$ |
\[-884736000 \,\,(\text{CM by }-43)\]
|
| $(-3 : -28 : 1)$ |
\[\frac{-18234932071051198464000}{48661191875666868481}\]
|
| $(-3 : 28 : 1)$ |
\[-262537412640768000 \,\,(\text{CM by }-163)\]
|
| $(3 : -28 : 1)$ |
\[0 \,\,(\text{CM by }-3)\]
|
| $(3 : 28 : 1)$ |
\[\frac{-35817550197738955933474532061609984000}
{2301619141096101839813550846721}
\]
|
|
| Comments on finding rational points |
The rational points on this curve were first resolved by Burcu Baran in a 2010 Journal
of Number Theory paper. The rank of the Jacobian is 2. We construct a family of
etale double covers, but one of these maps to an elliptic curve of rank 1. We
construct an etale four-fold cover over $\mathbb{Q}(\sqrt{2})$ that maps to an
elliptic curve over $\mathbb{Q}(\sqrt{2})$ and use elliptic curve Chabauty. |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + y = x^3 - 3014658660x + 150916472601529$, with conductor
$356257899$ |
| Generic density of odd order reductions |
$45875/86016$ |