Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 367 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 38\cdot 367 + 330\cdot 367^{2} + 114\cdot 367^{3} + 342\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 35 + 241\cdot 367 + 115\cdot 367^{2} + 290\cdot 367^{3} + 290\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 133\cdot 367 + 267\cdot 367^{2} + 38\cdot 367^{3} + 186\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 285 + 233\cdot 367 + 138\cdot 367^{2} + 94\cdot 367^{3} + 46\cdot 367^{4} +O\left(367^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 345 + 87\cdot 367 + 249\cdot 367^{2} + 195\cdot 367^{3} + 235\cdot 367^{4} +O\left(367^{ 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.
