Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 50\cdot 137 + 129\cdot 137^{2} + 107\cdot 137^{3} + 38\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 84\cdot 137 + 137^{2} + 72\cdot 137^{3} + 82\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 37 + 69\cdot 137 + 60\cdot 137^{3} + 20\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 + 84\cdot 137 + 11\cdot 137^{2} + 134\cdot 137^{3} + 95\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 77 + 70\cdot 137 + 5\cdot 137^{2} + 34\cdot 137^{3} + 132\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 115 + 88\cdot 137 + 109\cdot 137^{2} + 85\cdot 137^{3} + 97\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 120 + 49\cdot 137 + 119\cdot 137^{2} + 45\cdot 137^{3} + 25\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 136 + 50\cdot 137 + 33\cdot 137^{2} + 8\cdot 137^{3} + 55\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,8)(2,5,7,6)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,8)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,6)(7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,8)(2,5,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
