Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 439 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 + 92\cdot 439 + 391\cdot 439^{2} + 336\cdot 439^{3} + 239\cdot 439^{4} +O\left(439^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 85 + 106\cdot 439 + 335\cdot 439^{2} + 33\cdot 439^{3} + 144\cdot 439^{4} +O\left(439^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 343 + 27\cdot 439 + 371\cdot 439^{2} + 11\cdot 439^{3} + 241\cdot 439^{4} +O\left(439^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 418 + 28\cdot 439 + 104\cdot 439^{2} + 8\cdot 439^{3} + 10\cdot 439^{4} +O\left(439^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 419 + 183\cdot 439 + 115\cdot 439^{2} + 48\cdot 439^{3} + 243\cdot 439^{4} +O\left(439^{ 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 value |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
