| Curve name |
$X_{212}$ |
| 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 & 3 \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 $X_{212}$ minimally covers |
$X_{86}$, $X_{118}$, $X_{122}$ |
| Curves that minimally cover $X_{212}$ |
$X_{478}$, $X_{481}$, $X_{493}$, $X_{498}$, $X_{212a}$, $X_{212b}$, $X_{212c}$, $X_{212d}$, $X_{212e}$, $X_{212f}$, $X_{212g}$, $X_{212h}$, $X_{212i}$, $X_{212j}$, $X_{212k}$, $X_{212l}$ |
| Curves that minimally cover $X_{212}$ and have infinitely many rational
points. |
$X_{212a}$, $X_{212b}$, $X_{212c}$, $X_{212d}$, $X_{212e}$, $X_{212f}$, $X_{212g}$, $X_{212h}$, $X_{212i}$, $X_{212j}$, $X_{212k}$, $X_{212l}$ |
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{212}) = \mathbb{Q}(f_{212}), f_{86} =
\frac{f_{212}}{f_{212}^{2} - \frac{1}{8}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - 5391x + 285606$, with conductor $735$ |
| Generic density of odd order reductions |
$25/224$ |