Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 51 + 272\cdot 389 + 11\cdot 389^{2} + 150\cdot 389^{3} + 219\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 78 + 51\cdot 389 + 108\cdot 389^{2} + 47\cdot 389^{3} + 202\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 312 + 337\cdot 389 + 280\cdot 389^{2} + 341\cdot 389^{3} + 186\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 339 + 116\cdot 389 + 377\cdot 389^{2} + 238\cdot 389^{3} + 169\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
