Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 73\cdot 109 + 20\cdot 109^{2} + 49\cdot 109^{3} + 23\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 87\cdot 109 + 3\cdot 109^{2} + 27\cdot 109^{3} + 26\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 92\cdot 109 + 73\cdot 109^{2} + 28\cdot 109^{3} + 67\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 34\cdot 109 + 98\cdot 109^{2} + 104\cdot 109^{3} + 7\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 74\cdot 109 + 10\cdot 109^{2} + 4\cdot 109^{3} + 101\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 82 + 16\cdot 109 + 35\cdot 109^{2} + 80\cdot 109^{3} + 41\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 90 + 21\cdot 109 + 105\cdot 109^{2} + 81\cdot 109^{3} + 82\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 104 + 35\cdot 109 + 88\cdot 109^{2} + 59\cdot 109^{3} + 85\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,3,7)(2,8,5,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,4)(3,6)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,7)(2,8,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
