Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 46\cdot 97 + 47\cdot 97^{2} + 28\cdot 97^{3} + 14\cdot 97^{4} + 33\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 31 + 74\cdot 97 + 33\cdot 97^{2} + 66\cdot 97^{3} + 62\cdot 97^{4} + 76\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 13\cdot 97 + 10\cdot 97^{2} + 13\cdot 97^{3} + 90\cdot 97^{4} + 14\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 + 65\cdot 97 + 69\cdot 97^{2} + 69\cdot 97^{3} + 55\cdot 97^{4} + 51\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 51 + 31\cdot 97 + 27\cdot 97^{2} + 27\cdot 97^{3} + 41\cdot 97^{4} + 45\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 58 + 83\cdot 97 + 86\cdot 97^{2} + 83\cdot 97^{3} + 6\cdot 97^{4} + 82\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 66 + 22\cdot 97 + 63\cdot 97^{2} + 30\cdot 97^{3} + 34\cdot 97^{4} + 20\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 79 + 50\cdot 97 + 49\cdot 97^{2} + 68\cdot 97^{3} + 82\cdot 97^{4} + 63\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,8,4)(2,3,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
