Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 37\cdot 89 + 62\cdot 89^{2} + 26\cdot 89^{3} + 3\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 67\cdot 89 + 35\cdot 89^{2} + 37\cdot 89^{3} + 4\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 46 + 28\cdot 89 + 76\cdot 89^{2} + 12\cdot 89^{3} + 60\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 34\cdot 89 + 32\cdot 89^{2} + 72\cdot 89^{3} + 8\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 62 + 77\cdot 89 + 6\cdot 89^{2} + 66\cdot 89^{3} + 16\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 + 3\cdot 89 + 68\cdot 89^{2} + 87\cdot 89^{3} + 17\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 83 + 44\cdot 89 + 3\cdot 89^{2} + 12\cdot 89^{3} + 21\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 87 + 61\cdot 89 + 70\cdot 89^{2} + 40\cdot 89^{3} + 45\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,3,6)(2,5,7,4)$ |
| $(1,2)(3,7)(4,6)(5,8)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(2,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $1$ | $4$ | $(1,5,3,4)(2,8,7,6)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,5)(2,6,7,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,3,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,3,2)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,3,4)(2,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
