Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 137 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 62\cdot 137^{2} + 118\cdot 137^{3} + 65\cdot 137^{4} + 99\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 32\cdot 137 + 74\cdot 137^{2} + 53\cdot 137^{3} + 97\cdot 137^{4} + 130\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 134\cdot 137^{2} + 54\cdot 137^{3} + 136\cdot 137^{4} + 106\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 94 + 88\cdot 137 + 124\cdot 137^{2} + 90\cdot 137^{3} + 40\cdot 137^{4} + 41\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 108 + 84\cdot 137 + 29\cdot 137^{2} + 98\cdot 137^{3} + 7\cdot 137^{4} + 36\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 119 + 47\cdot 137 + 90\cdot 137^{2} + 9\cdot 137^{3} + 31\cdot 137^{4} + 26\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 130 + 3\cdot 137 + 20\cdot 137^{2} + 111\cdot 137^{3} + 98\cdot 137^{4} + 104\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 134 + 15\cdot 137 + 13\cdot 137^{2} + 11\cdot 137^{3} + 70\cdot 137^{4} + 2\cdot 137^{5} +O\left(137^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,5,4,7)(2,6,8,3)$ |
| $(1,8,5,3,4,2,7,6)$ |
| $(2,3)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $-2$ |
| $4$ | $2$ | $(2,3)(5,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,5,4,7)(2,6,8,3)$ | $0$ |
| $2$ | $8$ | $(1,8,5,3,4,2,7,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,7,8,4,6,5,2)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
