Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 110\cdot 137 + 99\cdot 137^{2} + 2\cdot 137^{3} + 120\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 130\cdot 137 + 62\cdot 137^{2} + 13\cdot 137^{3} + 132\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 33\cdot 137 + 61\cdot 137^{2} + 73\cdot 137^{3} + 91\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 59 + 75\cdot 137 + 88\cdot 137^{2} + 9\cdot 137^{3} + 27\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 + 61\cdot 137 + 48\cdot 137^{2} + 127\cdot 137^{3} + 109\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 90 + 103\cdot 137 + 75\cdot 137^{2} + 63\cdot 137^{3} + 45\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 119 + 6\cdot 137 + 74\cdot 137^{2} + 123\cdot 137^{3} + 4\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 125 + 26\cdot 137 + 37\cdot 137^{2} + 134\cdot 137^{3} + 16\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,4,6)(3,8,7,5)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,7)(3,6)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,2,4,6)(3,8,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
