Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 211 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 64 + 189\cdot 211 + 72\cdot 211^{2} + 166\cdot 211^{3} + 177\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 69 + 35\cdot 211 + 168\cdot 211^{2} + 177\cdot 211^{3} + 193\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 77 + 94\cdot 211 + 75\cdot 211^{2} + 65\cdot 211^{3} + 115\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 82 + 151\cdot 211 + 170\cdot 211^{2} + 76\cdot 211^{3} + 131\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 129 + 59\cdot 211 + 40\cdot 211^{2} + 134\cdot 211^{3} + 79\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 134 + 116\cdot 211 + 135\cdot 211^{2} + 145\cdot 211^{3} + 95\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 142 + 175\cdot 211 + 42\cdot 211^{2} + 33\cdot 211^{3} + 17\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 147 + 21\cdot 211 + 138\cdot 211^{2} + 44\cdot 211^{3} + 33\cdot 211^{4} +O\left(211^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,4,6)(2,5,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
