Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 47 + 65\cdot 149 + 90\cdot 149^{2} + 86\cdot 149^{3} + 31\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 + 50\cdot 149 + 123\cdot 149^{2} + 134\cdot 149^{3} + 63\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 90 + 112\cdot 149 + 116\cdot 149^{2} + 118\cdot 149^{3} + 105\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 96 + 43\cdot 149 + 41\cdot 149^{2} + 9\cdot 149^{3} + 112\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 98 + 51\cdot 149 + 22\cdot 149^{2} + 66\cdot 149^{3} + 24\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 106 + 130\cdot 149 + 61\cdot 149^{2} + 10\cdot 149^{3} + 29\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 118 + 26\cdot 149 + 43\cdot 149^{2} + 52\cdot 149^{3} + 124\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 144 + 114\cdot 149 + 96\cdot 149^{2} + 117\cdot 149^{3} + 104\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,6,3)(2,7,5,4)$ |
| $(1,5,6,2)(3,7,8,4)$ |
| $(1,6)(2,5)(3,8)(4,7)$ |
| $(3,8)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ |
| $2$ | $2$ | $(3,8)(4,7)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,5,6,2)(3,7,8,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,6,5)(3,4,8,7)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,6,3)(2,7,5,4)$ | $0$ |
| $2$ | $4$ | $(1,5,6,2)(3,4,8,7)$ | $0$ |
| $2$ | $4$ | $(1,4,6,7)(2,8,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
