Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 52\cdot 179 + 110\cdot 179^{2} + 159\cdot 179^{3} + 142\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 145\cdot 179 + 145\cdot 179^{2} + 99\cdot 179^{3} + 70\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 73 + 3\cdot 179 + 174\cdot 179^{2} + 176\cdot 179^{3} + 138\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 119 + 26\cdot 179 + 160\cdot 179^{2} + 91\cdot 179^{3} + 132\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 154 + 130\cdot 179 + 125\cdot 179^{2} + 8\cdot 179^{3} + 52\cdot 179^{4} +O\left(179^{ 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.
