| Curve name |
$X_{42a}$ |
| Index |
$24$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 0 \\ 14 & 15 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 10 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 14 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{42}$ |
| Curves that $X_{42a}$ minimally covers |
|
| Curves that minimally cover $X_{42a}$ |
|
| Curves that minimally cover $X_{42a}$ 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 |
$y^2 = x^3 + x^2 - 1533x + 29563$, with conductor $3200$ |
| Generic density of odd order reductions |
$85091/344064$ |