Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 60\cdot 149 + 19\cdot 149^{2} + 35\cdot 149^{3} + 26\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 127\cdot 149 + 25\cdot 149^{2} + 66\cdot 149^{3} + 55\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 61 + 130\cdot 149 + 146\cdot 149^{2} + 82\cdot 149^{3} + 137\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 78 + 67\cdot 149 + 144\cdot 149^{2} + 148\cdot 149^{3} + 28\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 86 + 72\cdot 149 + 76\cdot 149^{2} + 25\cdot 149^{3} + 116\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 99 + 6\cdot 149 + 84\cdot 149^{2} + 2\cdot 149^{3} + 70\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 123 + 40\cdot 149 + 115\cdot 149^{2} + 18\cdot 149^{3} + 103\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 127 + 90\cdot 149 + 132\cdot 149^{2} + 66\cdot 149^{3} + 58\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6)(2,4)(3,8)(5,7)$ |
| $(1,2,6,4)(3,7,8,5)$ |
| $(2,4)(3,8)$ |
| $(1,7)(2,8)(3,4)(5,6)$ |
| $(2,8,4,3)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,8)(5,7)$ |
$-4$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,3)(4,8)(5,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,6,4)(3,7,8,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,8,7,4)(2,6,3,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,7,8)(2,5,3,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,6,7)(2,4)$ |
$0$ |
| $4$ |
$4$ |
$(1,7,6,5)(2,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
