Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 149 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 119\cdot 149 + 126\cdot 149^{2} + 120\cdot 149^{3} + 28\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 46 + 37\cdot 149 + 132\cdot 149^{3} + 90\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 47 + 73\cdot 149 + 130\cdot 149^{2} + 90\cdot 149^{3} +O\left(149^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 82 + 47\cdot 149 + 85\cdot 149^{2} + 113\cdot 149^{3} + 61\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 86 + 32\cdot 149 + 38\cdot 149^{2} + 33\cdot 149^{3} + 148\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 89 + 79\cdot 149 + 85\cdot 149^{2} + 20\cdot 149^{3} + 98\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 84\cdot 149 + 73\cdot 149^{2} + 70\cdot 149^{3} + 95\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 125 + 121\cdot 149 + 55\cdot 149^{2} + 14\cdot 149^{3} + 72\cdot 149^{4} +O\left(149^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,4)(5,8)$ |
| $(1,3,6,7)(2,8,4,5)$ |
| $(1,6)(3,7)$ |
| $(1,5,3,2,6,8,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,7)(5,8)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(1,6)(3,7)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,3,6,7)(2,8,4,5)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,7,6,3)(2,5,4,8)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,3,6,7)(2,5,4,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,3,2,6,8,7,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,7,5,6,4,3,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,8,7,2,6,5,3,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,3,8,6,4,7,5)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
