Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 139 }$ to precision 5.
Roots:
$r_{ 1 }$ |
$=$ |
$ 48 + 52\cdot 139 + 37\cdot 139^{2} + 86\cdot 139^{3} + 30\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 2 }$ |
$=$ |
$ 55 + 24\cdot 139 + 70\cdot 139^{2} + 32\cdot 139^{3} + 93\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 3 }$ |
$=$ |
$ 74 + 93\cdot 139 + 61\cdot 139^{2} + 116\cdot 139^{3} + 51\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 4 }$ |
$=$ |
$ 77 + 83\cdot 139 + 15\cdot 139^{2} + 95\cdot 139^{3} + 98\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 5 }$ |
$=$ |
$ 90 + 116\cdot 139 + 20\cdot 139^{2} + 25\cdot 139^{3} + 118\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 6 }$ |
$=$ |
$ 93 + 94\cdot 139 + 130\cdot 139^{2} + 28\cdot 139^{3} + 62\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 7 }$ |
$=$ |
$ 122 + 72\cdot 139 + 124\cdot 139^{2} + 31\cdot 139^{3} + 123\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
$r_{ 8 }$ |
$=$ |
$ 136 + 17\cdot 139 + 95\cdot 139^{2} + 117\cdot 139^{4} +O\left(139^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
Cycle notation |
$(1,6,2,8)(3,5,4,7)$ |
$(3,7,4,5)$ |
$(3,4)(5,7)$ |
$(1,5)(2,7)(3,8)(4,6)$ |
$(1,2)(3,4)(5,7)(6,8)$ |
Character values on conjugacy classes
Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
$1$ | $1$ | $()$ | $2$ |
$1$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $-2$ |
$2$ | $2$ | $(3,4)(5,7)$ | $0$ |
$4$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $0$ |
$1$ | $4$ | $(1,6,2,8)(3,5,4,7)$ | $-2 \zeta_{4}$ |
$1$ | $4$ | $(1,8,2,6)(3,7,4,5)$ | $2 \zeta_{4}$ |
$2$ | $4$ | $(3,7,4,5)$ | $\zeta_{4} + 1$ |
$2$ | $4$ | $(3,5,4,7)$ | $-\zeta_{4} + 1$ |
$2$ | $4$ | $(1,2)(3,5,4,7)(6,8)$ | $-\zeta_{4} - 1$ |
$2$ | $4$ | $(1,2)(3,7,4,5)(6,8)$ | $\zeta_{4} - 1$ |
$2$ | $4$ | $(1,6,2,8)(3,7,4,5)$ | $0$ |
$4$ | $4$ | $(1,7,2,5)(3,8,4,6)$ | $0$ |
$4$ | $8$ | $(1,3,8,7,2,4,6,5)$ | $0$ |
$4$ | $8$ | $(1,7,6,3,2,5,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.