Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 + 54\cdot 137 + 7\cdot 137^{2} + 122\cdot 137^{3} + 33\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 55 + 86\cdot 137 + 97\cdot 137^{2} + 55\cdot 137^{3} + 35\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 71 + 34\cdot 137 + 42\cdot 137^{2} + 87\cdot 137^{3} + 72\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 86 + 3\cdot 137 + 105\cdot 137^{2} + 126\cdot 137^{3} + 73\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 89 + 44\cdot 137 + 119\cdot 137^{2} + 74\cdot 137^{3} + 93\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 103 + 27\cdot 137 + 87\cdot 137^{2} + 77\cdot 137^{3} + 114\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 120 + 98\cdot 137 + 126\cdot 137^{2} + 8\cdot 137^{3} + 132\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 135 + 60\cdot 137 + 99\cdot 137^{2} + 131\cdot 137^{3} + 128\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,4)(3,8)(5,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,7,3,2)(4,6,5,8)$ |
| $(1,3)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $1$ | $4$ | $(1,5,3,4)(2,6,7,8)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,5)(2,8,7,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,3,2)(4,6,5,8)$ | $0$ |
| $2$ | $4$ | $(1,6,3,8)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,4,3,5)(2,6,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
