Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 28\cdot 137 + 18\cdot 137^{2} + 10\cdot 137^{3} + 59\cdot 137^{4} + 86\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 59\cdot 137 + 64\cdot 137^{2} + 88\cdot 137^{3} + 18\cdot 137^{4} + 80\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 44\cdot 137 + 96\cdot 137^{2} + 4\cdot 137^{3} + 133\cdot 137^{4} + 134\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 69\cdot 137 + 56\cdot 137^{2} + 42\cdot 137^{3} + 64\cdot 137^{4} + 73\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 + 67\cdot 137 + 80\cdot 137^{2} + 94\cdot 137^{3} + 72\cdot 137^{4} + 63\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 110 + 92\cdot 137 + 40\cdot 137^{2} + 132\cdot 137^{3} + 3\cdot 137^{4} + 2\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 115 + 77\cdot 137 + 72\cdot 137^{2} + 48\cdot 137^{3} + 118\cdot 137^{4} + 56\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 124 + 108\cdot 137 + 118\cdot 137^{2} + 126\cdot 137^{3} + 77\cdot 137^{4} + 50\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,3,7,8,5,6,2)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,8)(2,5)(4,7)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,8)(2,5)(4,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,7,6,4,8,2,3,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
