Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 48\cdot 101 + 11\cdot 101^{2} + 56\cdot 101^{3} + 88\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 + 85\cdot 101 + 71\cdot 101^{2} + 10\cdot 101^{3} + 41\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 51 + 97\cdot 101 + 62\cdot 101^{2} + 56\cdot 101^{3} + 10\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 62 + 30\cdot 101 + 60\cdot 101^{2} + 14\cdot 101^{3} + 66\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 79 + 38\cdot 101 + 77\cdot 101^{2} + 10\cdot 101^{3} + 92\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 + 63\cdot 101 + 76\cdot 101^{2} + 15\cdot 101^{3} + 35\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 82 + 18\cdot 101 + 37\cdot 101^{2} + 7\cdot 101^{3} + 12\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 99 + 20\cdot 101 + 6\cdot 101^{2} + 30\cdot 101^{3} + 58\cdot 101^{4} +O\left(101^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,8,2)(3,4,7,6)$ |
| $(1,7,2,6,8,3,5,4)$ |
| $(3,7)(4,6)$ |
| $(1,8)(2,5)(3,7)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,5)(3,7)(4,6)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,7)(4,6)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,2,8,5)(3,4,7,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,5,8,2)(3,6,7,4)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,5,8,2)(3,4,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,2,6,8,3,5,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,5,7,8,4,2,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,2,3,8,4,5,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,5,6,8,7,2,4)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
