Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 30\cdot 101 + 100\cdot 101^{2} + 100\cdot 101^{3} + 11\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 84\cdot 101 + 84\cdot 101^{2} + 29\cdot 101^{3} + 14\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 52 + 100\cdot 101 + 29\cdot 101^{2} + 59\cdot 101^{3} + 94\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 63 + 87\cdot 101 + 78\cdot 101^{2} + 68\cdot 101^{3} + 54\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 66 + 84\cdot 101 + 93\cdot 101^{2} + 73\cdot 101^{3} + 40\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 89 + 57\cdot 101 + 74\cdot 101^{2} + 78\cdot 101^{3} + 79\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 82\cdot 101 + 58\cdot 101^{2} + 46\cdot 101^{3} + 70\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 77\cdot 101 + 84\cdot 101^{2} + 46\cdot 101^{3} + 37\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(6,8)$ |
| $(1,7,3,2)(4,8,5,6)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,8,3,6)(2,4,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,5)(2,8)(3,4)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,8,3,6)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,3,8)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,3,2)(4,8,5,6)$ | $0$ |
| $2$ | $4$ | $(1,6,3,8)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,4,3,5)(2,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
