Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 117\cdot 137 + 114\cdot 137^{2} + 91\cdot 137^{3} + 39\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 60 + 23\cdot 137 + 86\cdot 137^{2} + 129\cdot 137^{3} + 112\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 110\cdot 137 + 95\cdot 137^{2} + 86\cdot 137^{3} + 7\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 63\cdot 137 + 115\cdot 137^{2} + 25\cdot 137^{3} + 18\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 98 + 126\cdot 137 + 49\cdot 137^{2} + 135\cdot 137^{3} + 131\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 119 + 3\cdot 137 + 40\cdot 137^{2} + 111\cdot 137^{3} + 40\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 130 + 119\cdot 137 + 84\cdot 137^{2} + 69\cdot 137^{3} + 71\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 135 + 119\cdot 137 + 97\cdot 137^{2} + 34\cdot 137^{3} + 125\cdot 137^{4} +O\left(137^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,7)(5,6)$ |
| $(1,3)(4,7)(5,6)$ |
| $(1,3,4,7)(2,6,8,5)$ |
| $(1,6)(2,3)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,7)(5,6)$ | $-2$ |
| $4$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,3)(4,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,7)(2,6,8,5)$ | $0$ |
| $2$ | $8$ | $(1,5,7,8,4,6,3,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,8,3,5,4,2,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
