| Curve name |
$X_{66c}$ |
| Index |
$48$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 6 \\ 6 & 11 \end{matrix}\right],
\left[ \begin{matrix} 3 & 6 \\ 10 & 5 \end{matrix}\right],
\left[ \begin{matrix} 1 & 2 \\ 2 & 1 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 4 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{66}$ |
| Curves that $X_{66c}$ minimally covers |
|
| Curves that minimally cover $X_{66c}$ |
|
| Curves that minimally cover $X_{66c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -81t^{12} - 1944t^{11} - 25056t^{10} - 216000t^{9} - 1321920t^{8} -
5889024t^{7} - 19353600t^{6} - 47112192t^{5} - 84602880t^{4} - 110592000t^{3} -
102629376t^{2} - 63700992t - 21233664\]
\[B(t) = 3888t^{17} + 132192t^{16} + 2180736t^{15} + 23120640t^{14} +
176504832t^{13} + 1031546880t^{12} + 4790181888t^{11} + 18087100416t^{10} +
56251514880t^{9} + 144696803328t^{8} + 306571640832t^{7} + 528152002560t^{6} +
722963791872t^{5} + 757617131520t^{4} + 571666857984t^{3} + 277226717184t^{2} +
65229815808t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - 20425x - 1122000$, with conductor $5600$ |
| Generic density of odd order reductions |
$9249/57344$ |