Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 39\cdot 101 + 10\cdot 101^{2} + 76\cdot 101^{3} + 19\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 73\cdot 101 + 49\cdot 101^{2} + 95\cdot 101^{3} + 44\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 75\cdot 101 + 86\cdot 101^{2} + 2\cdot 101^{3} + 9\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 8\cdot 101 + 25\cdot 101^{2} + 22\cdot 101^{3} + 34\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 92\cdot 101 + 75\cdot 101^{2} + 78\cdot 101^{3} + 66\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 74 + 25\cdot 101 + 14\cdot 101^{2} + 98\cdot 101^{3} + 91\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 27\cdot 101 + 51\cdot 101^{2} + 5\cdot 101^{3} + 56\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 88 + 61\cdot 101 + 90\cdot 101^{2} + 24\cdot 101^{3} + 81\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,5)(3,4,8,7)$ |
| $(1,3)(2,7)(4,5)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,5)(3,4,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
