| Curve name |
$X_{101l}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 0 \\ 0 & 5 \end{matrix}\right],
\left[ \begin{matrix} 7 & 6 \\ 4 & 7 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 4 & 1 \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 $X_{101l}$ minimally covers |
|
| Curves that minimally cover $X_{101l}$ |
|
| Curves that minimally cover $X_{101l}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1769472t^{12} + 1769472t^{10} - 691200t^{8} + 131328t^{6} - 12528t^{4}
+ 648t^{2} - 27\]
\[B(t) = -905969664t^{18} + 1358954496t^{16} - 870580224t^{14} + 309657600t^{12}
- 66023424t^{10} + 8294400t^{8} - 508032t^{6} - 2592t^{4} + 1944t^{2} - 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 7059x - 227950$, with conductor $1008$ |
| Generic density of odd order reductions |
$25/224$ |