Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 179 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 26\cdot 179 + 87\cdot 179^{2} + 78\cdot 179^{3} + 94\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 + 171\cdot 179 + 37\cdot 179^{2} + 70\cdot 179^{3} + 166\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 94 + 13\cdot 179 + 16\cdot 179^{2} + 143\cdot 179^{3} + 67\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 95 + 136\cdot 179 + 74\cdot 179^{2} + 176\cdot 179^{3} + 26\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 138 + 178\cdot 179 + 50\cdot 179^{2} + 83\cdot 179^{3} + 89\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 144 + 124\cdot 179 + 101\cdot 179^{2} + 122\cdot 179^{3} + 45\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 152 + 42\cdot 179 + 43\cdot 179^{2} + 12\cdot 179^{3} + 47\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 177 + 21\cdot 179 + 125\cdot 179^{2} + 29\cdot 179^{3} + 178\cdot 179^{4} +O\left(179^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,5,7)$ |
| $(1,4,8,7)(2,6,5,3)$ |
| $(1,8)(2,5)(3,6)(4,7)$ |
| $(1,4,8,7)(2,3,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,6)(4,7)$ | $-2$ |
| $2$ | $2$ | $(2,5)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,4,8,7)(2,3,5,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,4)(2,6,5,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,8,7)(2,6,5,3)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,8,2)(3,4,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
