Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 379 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 114 + 121\cdot 379 + 351\cdot 379^{2} + 63\cdot 379^{3} + 106\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 173 + 190\cdot 379 + 278\cdot 379^{2} + 342\cdot 379^{3} + 238\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 258 + 267\cdot 379 + 365\cdot 379^{2} + 232\cdot 379^{3} + 255\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 263 + 316\cdot 379 + 262\cdot 379^{2} + 374\cdot 379^{3} + 125\cdot 379^{4} +O\left(379^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 329 + 240\cdot 379 + 257\cdot 379^{2} + 122\cdot 379^{3} + 31\cdot 379^{4} +O\left(379^{ 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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
