Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 112\cdot 127 + 49\cdot 127^{2} + 100\cdot 127^{3} + 113\cdot 127^{4} + 40\cdot 127^{5} + 75\cdot 127^{6} + 47\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 72\cdot 127 + 81\cdot 127^{2} + 105\cdot 127^{3} + 107\cdot 127^{4} + 121\cdot 127^{5} + 10\cdot 127^{6} + 49\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 14\cdot 127 + 15\cdot 127^{2} + 115\cdot 127^{3} + 19\cdot 127^{4} + 67\cdot 127^{5} + 28\cdot 127^{6} + 79\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 63 + 73\cdot 127 + 61\cdot 127^{2} + 25\cdot 127^{3} + 40\cdot 127^{4} + 37\cdot 127^{5} + 112\cdot 127^{6} + 105\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 64 + 53\cdot 127 + 65\cdot 127^{2} + 101\cdot 127^{3} + 86\cdot 127^{4} + 89\cdot 127^{5} + 14\cdot 127^{6} + 21\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 112\cdot 127 + 111\cdot 127^{2} + 11\cdot 127^{3} + 107\cdot 127^{4} + 59\cdot 127^{5} + 98\cdot 127^{6} + 47\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 116 + 54\cdot 127 + 45\cdot 127^{2} + 21\cdot 127^{3} + 19\cdot 127^{4} + 5\cdot 127^{5} + 116\cdot 127^{6} + 77\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 120 + 14\cdot 127 + 77\cdot 127^{2} + 26\cdot 127^{3} + 13\cdot 127^{4} + 86\cdot 127^{5} + 51\cdot 127^{6} + 79\cdot 127^{7} +O\left(127^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,8,5)(2,6,7,3)$ |
| $(2,3,7,6)$ |
| $(1,7,8,2)(3,4,6,5)$ |
| $(2,7)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(2,7)(3,6)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,8,4)(2,6,7,3)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(2,3,7,6)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(2,6,7,3)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,8)(2,6,7,3)(4,5)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,8)(2,3,7,6)(4,5)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $4$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,6,5,7,8,3,4,2)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,7,4,6,8,2,5,3)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
