Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 58\cdot 97 + 27\cdot 97^{2} + 34\cdot 97^{3} + 41\cdot 97^{4} + 88\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 36\cdot 97 + 71\cdot 97^{2} + 23\cdot 97^{3} + 22\cdot 97^{4} + 31\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 + 44\cdot 97 + 86\cdot 97^{2} + 49\cdot 97^{3} + 9\cdot 97^{4} + 80\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 + 22\cdot 97 + 33\cdot 97^{2} + 39\cdot 97^{3} + 87\cdot 97^{4} + 22\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 53 + 74\cdot 97 + 63\cdot 97^{2} + 57\cdot 97^{3} + 9\cdot 97^{4} + 74\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 52\cdot 97 + 10\cdot 97^{2} + 47\cdot 97^{3} + 87\cdot 97^{4} + 16\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 81 + 60\cdot 97 + 25\cdot 97^{2} + 73\cdot 97^{3} + 74\cdot 97^{4} + 65\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 93 + 38\cdot 97 + 69\cdot 97^{2} + 62\cdot 97^{3} + 55\cdot 97^{4} + 8\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,8)(4,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,7)(4,5)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,6,4)(2,3,5,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
