Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 89\cdot 127 + 15\cdot 127^{2} + 10\cdot 127^{3} + 42\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 68\cdot 127 + 77\cdot 127^{2} + 105\cdot 127^{3} + 44\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 116\cdot 127 + 32\cdot 127^{2} + 93\cdot 127^{3} + 99\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 65 + 126\cdot 127 + 60\cdot 127^{2} + 25\cdot 127^{3} + 72\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 19\cdot 127 + 118\cdot 127^{2} + 54\cdot 127^{3} + 38\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 79 + 84\cdot 127 + 16\cdot 127^{2} + 81\cdot 127^{3} + 18\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 99 + 7\cdot 127 + 103\cdot 127^{2} + 93\cdot 127^{3} + 36\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 119 + 122\cdot 127 + 82\cdot 127^{2} + 43\cdot 127^{3} + 28\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7)(2,6)(3,8)(4,5)$ |
| $(1,5)(6,8)$ |
| $(1,6,5,8)(2,4,3,7)$ |
| $(1,5)(2,3)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,6)(3,8)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,5)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,6,5,8)(2,4,3,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,8,5,6)(2,7,3,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,5,3)(4,8,7,6)$ | $0$ |
| $2$ | $4$ | $(1,7,5,4)(2,8,3,6)$ | $0$ |
| $2$ | $4$ | $(1,8,5,6)(2,4,3,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
