| Curve name |
$X_{205f}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{205}$ |
| Curves that $X_{205f}$ minimally covers |
|
| Curves that minimally cover $X_{205f}$ |
|
| Curves that minimally cover $X_{205f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{24} + 1566t^{20} - 405t^{16} - 3996t^{12} - 405t^{8} + 1566t^{4}
- 27\]
\[B(t) = 54t^{36} + 6966t^{32} - 35640t^{28} - 52920t^{24} + 67716t^{20} +
67716t^{16} - 52920t^{12} - 35640t^{8} + 6966t^{4} + 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 + 884334x + 498400200$, with conductor $8670$ |
| Generic density of odd order reductions |
$271/2688$ |