| Curve name |
$X_{99f}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 2 \\ 4 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{99}$ |
| Curves that $X_{99f}$ minimally covers |
|
| Curves that minimally cover $X_{99f}$ |
|
| Curves that minimally cover $X_{99f}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -27t^{16} - 270t^{14} - 1134t^{12} - 2592t^{10} - 3510t^{8} - 2916t^{6}
- 1539t^{4} - 540t^{2} - 108\]
\[B(t) = -54t^{24} - 810t^{22} - 5427t^{20} - 21411t^{18} - 55080t^{16} -
96228t^{14} - 114345t^{12} - 88209t^{10} - 37098t^{8} - 702t^{6} + 7452t^{4} +
3240t^{2} + 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 180300x - 29302000$, with conductor $14400$ |
| Generic density of odd order reductions |
$89/672$ |