Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 18\cdot 97 + 70\cdot 97^{2} + 57\cdot 97^{3} + 45\cdot 97^{4} + 2\cdot 97^{5} + 72\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 + 94\cdot 97 + 65\cdot 97^{2} + 57\cdot 97^{3} + 66\cdot 97^{4} + 24\cdot 97^{5} + 34\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 31 + 18\cdot 97 + 83\cdot 97^{2} + 74\cdot 97^{3} + 25\cdot 97^{4} + 28\cdot 97^{5} + 96\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 + 25\cdot 97 + 60\cdot 97^{2} + 45\cdot 97^{3} + 65\cdot 97^{4} + 89\cdot 97^{5} + 90\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 + 71\cdot 97 + 36\cdot 97^{2} + 51\cdot 97^{3} + 31\cdot 97^{4} + 7\cdot 97^{5} + 6\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 66 + 78\cdot 97 + 13\cdot 97^{2} + 22\cdot 97^{3} + 71\cdot 97^{4} + 68\cdot 97^{5} +O\left(97^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 67 + 2\cdot 97 + 31\cdot 97^{2} + 39\cdot 97^{3} + 30\cdot 97^{4} + 72\cdot 97^{5} + 62\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 78\cdot 97 + 26\cdot 97^{2} + 39\cdot 97^{3} + 51\cdot 97^{4} + 94\cdot 97^{5} + 24\cdot 97^{6} +O\left(97^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,8)(4,5)$ |
| $(1,3,8,6)(2,5,7,4)$ |
| $(2,7)(4,5)$ |
| $(1,5,8,4)(2,3,7,6)$ |
| $(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,8)(4,5)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$0$ |
| $2$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,8)(2,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,8,5)(2,3,7,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
