| Curve name |
$X_{42c}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{42}$ |
| Curves that $X_{42c}$ minimally covers |
|
| Curves that minimally cover $X_{42c}$ |
|
| Curves that minimally cover $X_{42c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 54t^{12} - 6048t^{10} - 41472t^{9} - 72576t^{8} + 331776t^{7} +
1880064t^{6} + 2654208t^{5} - 4644864t^{4} - 21233664t^{3} - 24772608t^{2} +
14155776\]
\[B(t) = -540t^{18} - 11664t^{17} - 95904t^{16} - 311040t^{15} + 414720t^{14} +
6469632t^{13} + 15704064t^{12} - 19906560t^{11} - 145981440t^{10} +
1167851520t^{8} + 1274019840t^{7} - 8040480768t^{6} - 26499612672t^{5} -
13589544960t^{4} + 81537269760t^{3} + 201125265408t^{2} + 195689447424t +
72477573120\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 383x + 3887$, with conductor $3200$ |
| Generic density of odd order reductions |
$85091/344064$ |