| Curve name |
$X_{223g}$ |
| Index |
$96$ |
| Level |
$16$ |
| Genus |
$0$ |
| Does the subgroup contain $-I$? |
No |
| Generating matrices |
$
\left[ \begin{matrix} 7 & 7 \\ 0 & 7 \end{matrix}\right],
\left[ \begin{matrix} 1 & 0 \\ 8 & 7 \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 $X_{223g}$ minimally covers |
|
| Curves that minimally cover $X_{223g}$ |
|
| Curves that minimally cover $X_{223g}$ and have infinitely many rational
points. |
|
| Model |
$\mathbb{P}^{1}$, a universal elliptic curve over an appropriate base is
given by
\[y^2 = x^3 + A(t)x + B(t), \text{ where}\]
\[A(t) = -7421703487488t^{24} - 18554258718720t^{22} - 16930761080832t^{20} -
6551972610048t^{18} - 809936879616t^{16} + 63417876480t^{14} + 15514730496t^{12}
+ 990904320t^{10} - 197738496t^{8} - 24993792t^{6} - 1009152t^{4} - 17280t^{2} -
108\]
\[B(t) = -7782220156096217088t^{36} - 29183325585360814080t^{34} -
44869363087492251648t^{32} - 35992768221945004032t^{30} -
15663238029017874432t^{28} - 3311623469745045504t^{26} -
128840772542791680t^{24} + 66973452271091712t^{22} + 10283697894850560t^{20} +
633396007010304t^{18} + 160682779607040t^{16} + 16350940495872t^{14} -
491488542720t^{12} - 197388140544t^{10} - 14587527168t^{8} - 523763712t^{6} -
10202112t^{4} - 103680t^{2} - 432\]
|
| Info about rational points |
| Comments on finding rational points |
None |
| Elliptic curve whose $2$-adic image is the subgroup |
$y^2 = x^3 - x^2 - 629249x - 177971295$, with conductor $55488$ |
| Generic density of odd order reductions |
$109/896$ |