| Curve name |
$X_{66b}$ |
| Index |
$48$ |
| Level |
$8$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 2 & 5 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 2 & 3 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{66b}$ minimally covers |
|
| Curves that minimally cover $X_{66b}$ |
|
| Curves that minimally cover $X_{66b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -81t^{16} - 2592t^{15} - 38016t^{14} - 338688t^{13} - 1942272t^{12} -
6359040t^{11} - 2875392t^{10} + 73875456t^{9} + 334209024t^{8} + 591003648t^{7}
- 184025088t^{6} - 3255828480t^{5} - 7955546112t^{4} - 11098128384t^{3} -
9965666304t^{2} - 5435817984t - 1358954496\]
\[B(t) = 3888t^{23} + 178848t^{22} + 3673728t^{21} + 44126208t^{20} +
337665024t^{19} + 1637922816t^{18} + 4167770112t^{17} - 3609722880t^{16} -
80372957184t^{15} - 350758895616t^{14} - 751756640256t^{13} +
6014053122048t^{11} + 22448569319424t^{10} + 41150954078208t^{9} +
14785424916480t^{8} - 136569491030016t^{7} - 429371638677504t^{6} -
708134880411648t^{5} - 740314922876928t^{4} - 493079425449984t^{3} -
192036577738752t^{2} - 33397665693696t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 40033x - 3078768$, with conductor $7840$ |
| Generic density of odd order reductions |
$149/896$ |