Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 109 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 70\cdot 109 + 107\cdot 109^{2} + 49\cdot 109^{3} + 21\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 13\cdot 109 + 104\cdot 109^{2} + 100\cdot 109^{3} + 59\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 12\cdot 109 + 102\cdot 109^{2} + 5\cdot 109^{3} + 30\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 97\cdot 109 + 38\cdot 109^{2} + 15\cdot 109^{3} + 7\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 + 36\cdot 109 + 29\cdot 109^{2} + 94\cdot 109^{3} + 15\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 98\cdot 109 + 85\cdot 109^{2} + 81\cdot 109^{3} + 11\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 55 + 13\cdot 109 + 13\cdot 109^{2} + 61\cdot 109^{3} + 106\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 75 + 94\cdot 109 + 63\cdot 109^{2} + 26\cdot 109^{3} + 74\cdot 109^{4} +O\left(109^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,3,2,7)(4,6,8,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,7)(4,8)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,8)(3,5)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,7)(4,6,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
