Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 61\cdot 97 + 89\cdot 97^{2} + 45\cdot 97^{3} + 31\cdot 97^{4} + 25\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 96\cdot 97 + 17\cdot 97^{2} + 49\cdot 97^{3} + 36\cdot 97^{4} + 44\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 52\cdot 97 + 72\cdot 97^{2} + 29\cdot 97^{3} + 91\cdot 97^{4} + 92\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 5\cdot 97 + 11\cdot 97^{2} + 62\cdot 97^{3} + 42\cdot 97^{4} + 10\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 65 + 91\cdot 97 + 85\cdot 97^{2} + 34\cdot 97^{3} + 54\cdot 97^{4} + 86\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 75 + 44\cdot 97 + 24\cdot 97^{2} + 67\cdot 97^{3} + 5\cdot 97^{4} + 4\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 81 + 79\cdot 97^{2} + 47\cdot 97^{3} + 60\cdot 97^{4} + 52\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 93 + 35\cdot 97 + 7\cdot 97^{2} + 51\cdot 97^{3} + 65\cdot 97^{4} + 71\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
