Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 15\cdot 127 + 60\cdot 127^{2} + 78\cdot 127^{3} + 55\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 51 + 34\cdot 127 + 51\cdot 127^{2} + 96\cdot 127^{3} + 108\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 64 + 34\cdot 127 + 104\cdot 127^{2} + 8\cdot 127^{3} + 65\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 89 + 3\cdot 127 + 48\cdot 127^{2} + 60\cdot 127^{3} + 53\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 + 72\cdot 127 + 58\cdot 127^{2} + 57\cdot 127^{3} + 105\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 94 + 22\cdot 127 + 57\cdot 127^{2} + 23\cdot 127^{3} + 60\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 104 + 113\cdot 127 + 85\cdot 127^{2} + 100\cdot 127^{3} + 109\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 111 + 83\cdot 127 + 42\cdot 127^{2} + 82\cdot 127^{3} + 76\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,8,7,6)$ |
| $(1,4)(2,3)(5,7)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,8)(2,6)(3,5)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)(5,7)(6,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,8)(2,6)(3,5)(4,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,3)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,4)(5,8,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,8,4,5,2,6,3,7)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,5,3,8,2,7,4,6)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
