Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 101\cdot 137 + 41\cdot 137^{2} + 51\cdot 137^{3} + 21\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 50 + 25\cdot 137 + 43\cdot 137^{2} + 94\cdot 137^{3} + 43\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 + 22\cdot 137 + 111\cdot 137^{2} + 70\cdot 137^{3} + 12\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 54 + 53\cdot 137 + 24\cdot 137^{2} + 23\cdot 137^{3} + 102\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 83 + 83\cdot 137 + 112\cdot 137^{2} + 113\cdot 137^{3} + 34\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 84 + 114\cdot 137 + 25\cdot 137^{2} + 66\cdot 137^{3} + 124\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 87 + 111\cdot 137 + 93\cdot 137^{2} + 42\cdot 137^{3} + 93\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 117 + 35\cdot 137 + 95\cdot 137^{2} + 85\cdot 137^{3} + 115\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,5)(2,3,4,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
