| Curve name |
$X_{42d}$ |
| Index |
$24$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 2 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{42}$ |
| Curves that $X_{42d}$ minimally covers |
|
| Curves that minimally cover $X_{42d}$ |
|
| Curves that minimally cover $X_{42d}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 54t^{8} - 432t^{7} - 4320t^{6} + 3456t^{5} + 62208t^{4} + 27648t^{3} -
276480t^{2} - 221184t + 221184\]
\[B(t) = 540t^{12} + 5184t^{11} - 5184t^{10} - 138240t^{9} - 20736t^{8} +
1658880t^{7} - 13271040t^{5} + 1327104t^{4} + 70778880t^{3} + 21233664t^{2} -
169869312t - 141557760\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 15x - 25$, with conductor $640$ |
| Generic density of odd order reductions |
$401/1792$ |