Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 53\cdot 101 + 89\cdot 101^{2} + 101^{3} + 95\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 39\cdot 101 + 71\cdot 101^{2} + 24\cdot 101^{3} + 29\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 42\cdot 101 + 10\cdot 101^{2} + 89\cdot 101^{3} + 94\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 34\cdot 101 + 93\cdot 101^{2} + 52\cdot 101^{3} + 95\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 56 + 87\cdot 101 + 7\cdot 101^{2} + 76\cdot 101^{3} + 3\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 + 74\cdot 101 + 2\cdot 101^{2} + 68\cdot 101^{3} + 29\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 37\cdot 101 + 90\cdot 101^{2} + 84\cdot 101^{3} + 7\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 97 + 34\cdot 101 + 38\cdot 101^{2} + 6\cdot 101^{3} + 48\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(4,5)$ |
| $(1,8,2,6)(3,4,7,5)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,7,2,3)(4,6,5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,7)(4,5)(6,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,6)(4,5)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,2,6)(3,4,7,5)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,7,2,3)(4,6,5,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,6,4,2,3,8,5)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,6,5,2,7,8,4)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
