| Curve name |
$X_{229}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 5 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{86}$ |
| Curves that $X_{229}$ minimally covers |
$X_{86}$, $X_{117}$, $X_{120}$ |
| Curves that minimally cover $X_{229}$ |
$X_{467}$, $X_{476}$, $X_{229a}$, $X_{229b}$, $X_{229c}$, $X_{229d}$, $X_{229e}$, $X_{229f}$, $X_{229g}$, $X_{229h}$ |
| Curves that minimally cover $X_{229}$ and have infinitely many rational
points. |
$X_{229a}$, $X_{229b}$, $X_{229c}$, $X_{229d}$, $X_{229e}$, $X_{229f}$, $X_{229g}$, $X_{229h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{229}) = \mathbb{Q}(f_{229}), f_{86} =
\frac{f_{229}^{2} - \frac{1}{2}}{f_{229}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 + 2034x + 60264$, with conductor $306$ |
| Generic density of odd order reductions |
$635/5376$ |