Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 48\cdot 103 + 97\cdot 103^{2} + 16\cdot 103^{3} + 12\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 92\cdot 103 + 71\cdot 103^{2} + 47\cdot 103^{3} + 51\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 36 + 41\cdot 103 + 96\cdot 103^{2} + 17\cdot 103^{3} + 93\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 91\cdot 103 + 51\cdot 103^{2} + 43\cdot 103^{3} + 42\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 60 + 32\cdot 103 + 55\cdot 103^{2} + 70\cdot 103^{3} + 53\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 31\cdot 103 + 10\cdot 103^{2} + 44\cdot 103^{3} + 78\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 73 + 92\cdot 103 + 26\cdot 103^{2} + 44\cdot 103^{3} + 58\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 100 + 84\cdot 103 + 103^{2} + 24\cdot 103^{3} + 22\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,3,2,8,7,6,5)$ |
| $(1,8)(2,7)(4,5)$ |
| $(1,8)(2,5)(3,6)(4,7)$ |
| $(1,6,8,3)(2,4,5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,5)(3,6)(4,7)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,7)(4,5)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,7,5,4)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,5,8,2)(3,7,6,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,3,2,8,7,6,5)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,7,3,5,8,4,6,2)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
