Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 51\cdot 97 + 42\cdot 97^{2} + 18\cdot 97^{3} + 26\cdot 97^{4} + 34\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 63\cdot 97 + 36\cdot 97^{2} + 29\cdot 97^{4} + 64\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 19\cdot 97 + 60\cdot 97^{2} + 77\cdot 97^{3} + 51\cdot 97^{4} + 8\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 60\cdot 97 + 54\cdot 97^{2} + 87\cdot 97^{4} + 86\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 + 36\cdot 97 + 42\cdot 97^{2} + 96\cdot 97^{3} + 9\cdot 97^{4} + 10\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 67 + 77\cdot 97 + 36\cdot 97^{2} + 19\cdot 97^{3} + 45\cdot 97^{4} + 88\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 72 + 33\cdot 97 + 60\cdot 97^{2} + 96\cdot 97^{3} + 67\cdot 97^{4} + 32\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 45\cdot 97 + 54\cdot 97^{2} + 78\cdot 97^{3} + 70\cdot 97^{4} + 62\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,8,7,6)$ |
| $(1,5)(2,6)(3,7)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,7)(3,8)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,4)(5,8,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
