| Curve name |
$X_{196}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
Yes |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 2 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{64}$ |
| Curves that $X_{196}$ minimally covers |
$X_{64}$ |
| Curves that minimally cover $X_{196}$ |
$X_{443}$, $X_{444}$, $X_{457}$, $X_{461}$ |
| Curves that minimally cover $X_{196}$ and have infinitely many rational
points. |
|
| Model |
\[\mathbb{P}^{1}, \mathbb{Q}(X_{196}) = \mathbb{Q}(f_{196}), f_{64} =
\frac{f_{196}^{2} + \frac{1}{2}f_{196} + \frac{1}{8}}{f_{196}}\] |
| Info about rational points |
None |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + x^2 - 6738x - 209880$, with conductor $294$ |
| Generic density of odd order reductions |
$269/1344$ |