| Curve name |
$X_{223a}$ |
| 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} 3 & 0 \\ 8 & 7 \end{matrix}\right],
\left[ \begin{matrix} 7 & 0 \\ 8 & 1 \end{matrix}\right]$ |
| Images in lower levels |
|
| Meaning/Special name |
|
| Chosen covering |
$X_{223}$ |
| Curves that $X_{223a}$ minimally covers |
|
| Curves that minimally cover $X_{223a}$ |
|
| Curves that minimally cover $X_{223a}$ 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 |
$109/896$ |