Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 101 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 80\cdot 101 + 6\cdot 101^{2} + 81\cdot 101^{3} + 82\cdot 101^{4} + 64\cdot 101^{5} + 26\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 93\cdot 101 + 60\cdot 101^{2} + 4\cdot 101^{3} + 88\cdot 101^{4} + 2\cdot 101^{5} + 33\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 43\cdot 101 + 60\cdot 101^{2} + 10\cdot 101^{3} + 95\cdot 101^{4} + 97\cdot 101^{5} + 80\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 + 17\cdot 101 + 7\cdot 101^{2} + 62\cdot 101^{3} + 97\cdot 101^{4} + 64\cdot 101^{5} + 22\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 83\cdot 101 + 93\cdot 101^{2} + 38\cdot 101^{3} + 3\cdot 101^{4} + 36\cdot 101^{5} + 78\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 57\cdot 101 + 40\cdot 101^{2} + 90\cdot 101^{3} + 5\cdot 101^{4} + 3\cdot 101^{5} + 20\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 74 + 7\cdot 101 + 40\cdot 101^{2} + 96\cdot 101^{3} + 12\cdot 101^{4} + 98\cdot 101^{5} + 67\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 96 + 20\cdot 101 + 94\cdot 101^{2} + 19\cdot 101^{3} + 18\cdot 101^{4} + 36\cdot 101^{5} + 74\cdot 101^{6} +O\left(101^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(2,3,7,6)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(2,7)(3,6)$ |
| $(1,5,8,4)(2,6,7,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(2,7)(3,6)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,5,8,4)(2,6,7,3)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $2$ |
$4$ |
$(2,3,7,6)$ |
$\zeta_{4} + 1$ |
$-\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(2,6,7,3)$ |
$-\zeta_{4} + 1$ |
$\zeta_{4} + 1$ |
| $2$ |
$4$ |
$(1,8)(2,6,7,3)(4,5)$ |
$-\zeta_{4} - 1$ |
$\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,8)(2,3,7,6)(4,5)$ |
$\zeta_{4} - 1$ |
$-\zeta_{4} - 1$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,3,4,7,8,6,5,2)$ |
$0$ |
$0$ |
| $4$ |
$8$ |
$(1,7,5,3,8,2,4,6)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
