Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 55\cdot 89 + 59\cdot 89^{2} + 79\cdot 89^{3} + 61\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 + 47\cdot 89 + 30\cdot 89^{2} + 69\cdot 89^{3} + 6\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 12\cdot 89 + 33\cdot 89^{2} + 15\cdot 89^{3} + 2\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 86\cdot 89 + 10\cdot 89^{2} + 3\cdot 89^{3} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 73 + 33\cdot 89 + 22\cdot 89^{2} + 63\cdot 89^{3} + 33\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 63\cdot 89 + 49\cdot 89^{2} + 9\cdot 89^{3} + 17\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 74\cdot 89 + 42\cdot 89^{2} + 3\cdot 89^{3} + 55\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 83 + 70\cdot 89 + 17\cdot 89^{2} + 23\cdot 89^{3} + 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(4,6)$ |
| $(1,4,2,6)(3,8,7,5)$ |
| $(1,3,4,5,2,7,6,8)$ |
| $(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,6)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,4,2,6)(3,5,7,8)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,2,4)(3,8,7,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,4,2,6)(3,8,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,4,5,2,7,6,8)$ | $0$ |
| $2$ | $8$ | $(1,5,6,3,2,8,4,7)$ | $0$ |
| $2$ | $8$ | $(1,7,6,5,2,3,4,8)$ | $0$ |
| $2$ | $8$ | $(1,5,4,7,2,8,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
