Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 18\cdot 127 + 121\cdot 127^{2} + 22\cdot 127^{3} + 60\cdot 127^{4} + 70\cdot 127^{5} + 26\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 + 116\cdot 127 + 49\cdot 127^{2} + 123\cdot 127^{3} + 65\cdot 127^{4} + 93\cdot 127^{5} + 91\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 110\cdot 127 + 28\cdot 127^{2} + 59\cdot 127^{3} + 111\cdot 127^{4} + 47\cdot 127^{5} + 121\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 + 17\cdot 127 + 8\cdot 127^{2} + 24\cdot 127^{3} + 123\cdot 127^{4} + 64\cdot 127^{5} + 83\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 76 + 109\cdot 127 + 118\cdot 127^{2} + 102\cdot 127^{3} + 3\cdot 127^{4} + 62\cdot 127^{5} + 43\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 + 16\cdot 127 + 98\cdot 127^{2} + 67\cdot 127^{3} + 15\cdot 127^{4} + 79\cdot 127^{5} + 5\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 95 + 10\cdot 127 + 77\cdot 127^{2} + 3\cdot 127^{3} + 61\cdot 127^{4} + 33\cdot 127^{5} + 35\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 114 + 108\cdot 127 + 5\cdot 127^{2} + 104\cdot 127^{3} + 66\cdot 127^{4} + 56\cdot 127^{5} + 100\cdot 127^{6} +O\left(127^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,8,5)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,8)(4,5)$ |
| $(1,5,8,4)(2,6,7,3)$ |
| $(1,3,5,2,8,6,4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,8,4)(2,6,7,3)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,4,8,5)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,5,8,4)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,7)(3,6)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,7)(3,6)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,6,8,3)(2,5,7,4)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,3,5,2,8,6,4,7)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,2,4,3,8,7,5,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
