| Curve name |
$X_{227b}$ |
| Index |
$96$ |
| Level |
$32$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 5 & 0 \\ 16 & 3 \end{matrix}\right],
\left[ \begin{matrix} 7 & 7 \\ 16 & 1 \end{matrix}\right],
\left[ \begin{matrix} 5 & 0 \\ 0 & 1 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 16 & 7 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{227}$ |
| Curves that $X_{227b}$ minimally covers |
|
| Curves that minimally cover $X_{227b}$ |
|
| Curves that minimally cover $X_{227b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1811939328t^{24} + 54358179840t^{22} - 60926459904t^{20} +
16986931200t^{18} - 106168320t^{16} - 1274019840t^{14} + 490143744t^{12} -
79626240t^{10} - 414720t^{8} + 4147200t^{6} - 929664t^{4} + 51840t^{2} - 108\]
\[B(t) = 29686813949952t^{36} + 1870269278846976t^{34} - 7720427052859392t^{32}
+ 6779726135820288t^{30} - 2414372915773440t^{28} + 102280351186944t^{26} +
211663504539648t^{24} - 94061394395136t^{22} + 20658826248192t^{20} -
1291176640512t^{16} + 367427321856t^{14} - 51675660288t^{12} - 1560674304t^{10}
+ 2302525440t^{8} - 404103168t^{6} + 28760832t^{4} - 435456t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 + x^2 + 37359967x - 28559579937$, with conductor $196800$ |
| Generic density of odd order reductions |
$299/2688$ |