Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 81\cdot 137 + 51\cdot 137^{2} + 52\cdot 137^{3} + 72\cdot 137^{4} + 48\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 + 33\cdot 137 + 49\cdot 137^{2} + 69\cdot 137^{4} + 78\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 + 137 + 95\cdot 137^{2} + 113\cdot 137^{3} + 103\cdot 137^{4} + 83\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 63\cdot 137 + 29\cdot 137^{2} + 11\cdot 137^{3} + 34\cdot 137^{4} + 62\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 91 + 73\cdot 137 + 107\cdot 137^{2} + 125\cdot 137^{3} + 102\cdot 137^{4} + 74\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 93 + 135\cdot 137 + 41\cdot 137^{2} + 23\cdot 137^{3} + 33\cdot 137^{4} + 53\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 + 103\cdot 137 + 87\cdot 137^{2} + 136\cdot 137^{3} + 67\cdot 137^{4} + 58\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 110 + 55\cdot 137 + 85\cdot 137^{2} + 84\cdot 137^{3} + 64\cdot 137^{4} + 88\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,6)(3,8)(4,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,6)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
