Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 409 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 238 + 165\cdot 409 + 359\cdot 409^{2} + 111\cdot 409^{3} + 367\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 294 + 378\cdot 409 + 316\cdot 409^{2} + 407\cdot 409^{3} + 351\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 352 + 50\cdot 409 + 2\cdot 409^{2} + 255\cdot 409^{3} + 121\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 356 + 109\cdot 409 + 310\cdot 409^{2} + 341\cdot 409^{3} + 45\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 397 + 112\cdot 409 + 238\cdot 409^{2} + 110\cdot 409^{3} + 340\cdot 409^{4} +O\left(409^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
