Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 389 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 168\cdot 389 + 96\cdot 389^{2} + 286\cdot 389^{3} + 371\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 128 + 323\cdot 389 + 119\cdot 389^{2} + 197\cdot 389^{3} + 32\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 261 + 65\cdot 389 + 269\cdot 389^{2} + 191\cdot 389^{3} + 356\cdot 389^{4} +O\left(389^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 362 + 220\cdot 389 + 292\cdot 389^{2} + 102\cdot 389^{3} + 17\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.
