Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 58\cdot 137 + 98\cdot 137^{2} + 9\cdot 137^{3} + 13\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 13\cdot 137 + 118\cdot 137^{2} + 51\cdot 137^{3} + 72\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 38 + 59\cdot 137 + 112\cdot 137^{2} + 53\cdot 137^{3} + 49\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 49 + 6\cdot 137 + 82\cdot 137^{2} + 21\cdot 137^{3} + 2\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 89 + 130\cdot 137 + 54\cdot 137^{2} + 115\cdot 137^{3} + 134\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 100 + 77\cdot 137 + 24\cdot 137^{2} + 83\cdot 137^{3} + 87\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 104 + 123\cdot 137 + 18\cdot 137^{2} + 85\cdot 137^{3} + 64\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 120 + 78\cdot 137 + 38\cdot 137^{2} + 127\cdot 137^{3} + 123\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,2,6)(3,8,4,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,8,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
