Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 21\cdot 127 + 116\cdot 127^{2} + 9\cdot 127^{3} + 72\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 + 122\cdot 127 + 27\cdot 127^{2} + 20\cdot 127^{3} + 10\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 13\cdot 127 + 43\cdot 127^{2} + 33\cdot 127^{3} + 40\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 47 + 79\cdot 127 + 111\cdot 127^{2} + 84\cdot 127^{3} + 123\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 25\cdot 127 + 117\cdot 127^{2} + 61\cdot 127^{3} + 2\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 + 37\cdot 127 + 101\cdot 127^{2} + 115\cdot 127^{3} + 67\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 100 + 50\cdot 127 + 105\cdot 127^{2} + 107\cdot 127^{3} + 92\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 123 + 30\cdot 127 + 12\cdot 127^{2} + 74\cdot 127^{3} + 98\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,3)(4,7)(6,8)$ |
| $(1,6,5,8)(2,7,3,4)$ |
| $(1,4,8,3,5,7,6,2)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,7)(6,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,3)(5,6)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,8,5,6)(2,4,3,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,4,8,3,5,7,6,2)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,3,6,4,5,2,8,7)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
