| Curve name |
$X_{225}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{85}$ |
| Curves that $X_{225}$ minimally covers |
$X_{85}$, $X_{118}$, $X_{121}$ |
| Curves that minimally cover $X_{225}$ |
$X_{471}$, $X_{482}$, $X_{495}$, $X_{497}$, $X_{225a}$, $X_{225b}$, $X_{225c}$, $X_{225d}$, $X_{225e}$, $X_{225f}$, $X_{225g}$, $X_{225h}$, $X_{225i}$, $X_{225j}$, $X_{225k}$, $X_{225l}$ |
| Curves that minimally cover $X_{225}$ and have infinitely many rational
points. |
$X_{225a}$, $X_{225b}$, $X_{225c}$, $X_{225d}$, $X_{225e}$, $X_{225f}$, $X_{225g}$, $X_{225h}$, $X_{225i}$, $X_{225j}$, $X_{225k}$, $X_{225l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{225}) = \mathbb{Q}(f_{225}), f_{85} =
\frac{f_{225}}{f_{225}^{2} + \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 105841x + 13244636$, with conductor $735$ |
| Generic density of odd order reductions |
$25/224$ |