Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 26 + 2\cdot 101 + 88\cdot 101^{2} + 18\cdot 101^{3} + 69\cdot 101^{4} + 61\cdot 101^{5} + 43\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 + 31\cdot 101 + 88\cdot 101^{2} + 34\cdot 101^{3} + 61\cdot 101^{4} + 98\cdot 101^{5} + 39\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 + 18\cdot 101 + 39\cdot 101^{2} + 45\cdot 101^{3} + 36\cdot 101^{4} + 100\cdot 101^{5} + 63\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 53 + 42\cdot 101 + 80\cdot 101^{2} + 25\cdot 101^{3} + 77\cdot 101^{4} + 37\cdot 101^{5} + 28\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 37\cdot 101 + 95\cdot 101^{2} + 10\cdot 101^{3} + 19\cdot 101^{4} + 2\cdot 101^{5} + 66\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 86 + 79\cdot 101 + 50\cdot 101^{2} + 84\cdot 101^{3} + 70\cdot 101^{4} + 22\cdot 101^{5} + 10\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 87 + 24\cdot 101 + 46\cdot 101^{2} + 21\cdot 101^{3} + 95\cdot 101^{4} + 3\cdot 101^{5} + 90\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 + 65\cdot 101 + 16\cdot 101^{2} + 61\cdot 101^{3} + 75\cdot 101^{4} + 76\cdot 101^{5} + 61\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,5)(6,8)$ |
| $(1,3,8,7,4,5,6,2)$ |
| $(2,7)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,5)(6,8)$ |
$-4$ |
| $2$ |
$2$ |
$(2,7)(3,5)$ |
$0$ |
| $4$ |
$2$ |
$(3,5)(6,8)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,3)(4,6)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,4,6)(2,3,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,4,6)(2,5,7,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,8,7,4,5,6,2)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,6,3,4,2,8,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,6,7,4,5,8,2)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,8,3,4,2,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
