Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 127 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 73 + 4\cdot 127 + 96\cdot 127^{2} + 36\cdot 127^{3} + 121\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 79 + 89\cdot 127 + 11\cdot 127^{2} + 17\cdot 127^{3} + 50\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 85 + 116\cdot 127 + 66\cdot 127^{2} + 43\cdot 127^{3} + 66\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 + 7\cdot 127 + 30\cdot 127^{2} + 31\cdot 127^{3} + 40\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 + 33\cdot 127 + 64\cdot 127^{2} + 105\cdot 127^{3} + 96\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 105 + 88\cdot 127 + 57\cdot 127^{2} + 85\cdot 127^{3} + 70\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 115 + 122\cdot 127 + 20\cdot 127^{2} + 100\cdot 127^{3} + 66\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 120 + 43\cdot 127 + 33\cdot 127^{2} + 88\cdot 127^{3} + 122\cdot 127^{4} +O\left(127^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,7)$ |
| $(1,4)(2,3)(5,6)(7,8)$ |
| $(1,6)(2,7)(3,8)(4,5)$ |
| $(1,2,6,7)(3,4,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,7,6,2)(3,4,8,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,6,7)(3,5,8,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,6,7)(3,4,8,5)$ | $0$ |
| $2$ | $4$ | $(1,4,6,5)(2,3,7,8)$ | $0$ |
| $2$ | $4$ | $(1,8,6,3)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
