| Curve name |
$X_{42b}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 6 & 7 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 6 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{42}$ |
| Curves that $X_{42b}$ minimally covers |
|
| Curves that minimally cover $X_{42b}$ |
|
| Curves that minimally cover $X_{42b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 216t^{8} - 1728t^{7} - 17280t^{6} + 13824t^{5} + 248832t^{4} +
110592t^{3} - 1105920t^{2} - 884736t + 884736\]
\[B(t) = 4320t^{12} + 41472t^{11} - 41472t^{10} - 1105920t^{9} - 165888t^{8} +
13271040t^{7} - 106168320t^{5} + 10616832t^{4} + 566231040t^{3} + 169869312t^{2}
- 1358954496t - 1132462080\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 61x + 261$, with conductor $640$ |
| Generic density of odd order reductions |
$1427/5376$ |