| Curve name |
$X_{223b}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 1 & 1 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right],
\left[ \begin{matrix} 3 & 0 \\ 0 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{223}$ |
| Curves that $X_{223b}$ minimally covers |
|
| Curves that minimally cover $X_{223b}$ |
|
| Curves that minimally cover $X_{223b}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -1855425871872t^{24} - 4638564679680t^{22} - 4232690270208t^{20} -
1637993152512t^{18} - 202484219904t^{16} + 15854469120t^{14} + 3878682624t^{12}
+ 247726080t^{10} - 49434624t^{8} - 6248448t^{6} - 252288t^{4} - 4320t^{2} -
27\]
\[B(t) = 972777519512027136t^{36} + 3647915698170101760t^{34} +
5608670385936531456t^{32} + 4499096027743125504t^{30} +
1957904753627234304t^{28} + 413952933718130688t^{26} + 16105096567848960t^{24} -
8371681533886464t^{22} - 1285462236856320t^{20} - 79174500876288t^{18} -
20085347450880t^{16} - 2043867561984t^{14} + 61436067840t^{12} +
24673517568t^{10} + 1823440896t^{8} + 65470464t^{6} + 1275264t^{4} + 12960t^{2}
+ 54\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 + xy + y = x^3 + x^2 - 9832x + 343913$, with conductor $1734$ |
| Generic density of odd order reductions |
$299/2688$ |