| Curve name |
$X_{231c}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 3 & 3 \\ 4 & 13 \end{matrix}\right],
\left[ \begin{matrix} 3 & 3 \\ 0 & 13 \end{matrix}\right],
\left[ \begin{matrix} 3 & 9 \\ 4 & 5 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{231}$ |
| Curves that $X_{231c}$ minimally covers |
|
| Curves that minimally cover $X_{231c}$ |
|
| Curves that minimally cover $X_{231c}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = 324t^{16} + 5184t^{15} + 13824t^{14} - 169344t^{13} - 1294272t^{12} -
2951424t^{11} + 1935360t^{10} + 23459328t^{9} + 50239872t^{8} + 46918656t^{7} +
7741440t^{6} - 23611392t^{5} - 20708352t^{4} - 5419008t^{3} + 884736t^{2} +
663552t + 82944\]
\[B(t) = 31104t^{23} + 715392t^{22} + 6130944t^{21} + 18579456t^{20} -
53125632t^{19} - 571802112t^{18} - 1321712640t^{17} + 2526806016t^{16} +
22755962880t^{15} + 60751171584t^{14} + 74833403904t^{13} - 149666807808t^{11} -
243004686336t^{10} - 182047703040t^{9} - 40428896256t^{8} + 42294804480t^{7} +
36595335168t^{6} + 6800080896t^{5} - 4756340736t^{4} - 3139043328t^{3} -
732561408t^{2} - 63700992t\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + 960388x + 520169216$, with conductor $84320$ |
| Generic density of odd order reductions |
$9827/86016$ |