| Curve name |
$X_{101p}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 0 & 3 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{101}$ |
| Curves that $X_{101p}$ minimally covers |
|
| Curves that minimally cover $X_{101p}$ |
|
| Curves that minimally cover $X_{101p}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7077888t^{12} + 7077888t^{10} - 2764800t^{8} + 525312t^{6} -
50112t^{4} + 2592t^{2} - 108\]
\[B(t) = -7247757312t^{18} + 10871635968t^{16} - 6964641792t^{14} +
2477260800t^{12} - 528187392t^{10} + 66355200t^{8} - 4064256t^{6} - 20736t^{4} +
15552t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy = x^3 - x^2 - 63720x - 5613300$, with conductor $2070$ |
| Generic density of odd order reductions |
$193/1792$ |