| Curve name |
$X_{226}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 9 & 9 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 15 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{79}$ |
| Curves that $X_{226}$ minimally covers |
$X_{79}$ |
| Curves that minimally cover $X_{226}$ |
$X_{226a}$, $X_{226b}$, $X_{226c}$, $X_{226d}$, $X_{226e}$, $X_{226f}$, $X_{226g}$, $X_{226h}$ |
| Curves that minimally cover $X_{226}$ and have infinitely many rational
points. |
$X_{226a}$, $X_{226b}$, $X_{226c}$, $X_{226d}$, $X_{226e}$, $X_{226f}$, $X_{226g}$, $X_{226h}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{226}) = \mathbb{Q}(f_{226}), f_{79} =
\frac{4f_{226} + 4}{f_{226}^{2} - 2}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 55862675x + 151569825750$, with conductor $421600$ |
| Generic density of odd order reductions |
$9249/57344$ |