Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 34\cdot 89 + 41\cdot 89^{2} + 48\cdot 89^{3} + 13\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 15\cdot 89 + 70\cdot 89^{2} + 32\cdot 89^{3} + 60\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 55\cdot 89 + 15\cdot 89^{2} + 35\cdot 89^{3} + 70\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 58\cdot 89 + 46\cdot 89^{2} + 69\cdot 89^{3} + 85\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 47 + 71\cdot 89 + 65\cdot 89^{2} + 17\cdot 89^{3} + 76\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 + 64\cdot 89 + 82\cdot 89^{2} + 65\cdot 89^{3} + 34\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 41\cdot 89^{2} + 75\cdot 89^{3} + 88\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 79 + 55\cdot 89 + 81\cdot 89^{2} + 10\cdot 89^{3} + 15\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,2)(4,8)(6,7)$ |
| $(1,2,4,8)(3,6,5,7)$ |
| $(1,7,4,6)(2,5,8,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,8)(3,5)(6,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,2)(4,8)(6,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,2,4,8)(3,6,5,7)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,7,4,6)(2,5,8,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,2,5,4,7,8,3)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,7,2,3,4,6,8,5)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
