Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 80\cdot 97 + 37\cdot 97^{2} + 72\cdot 97^{3} + 43\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 10 + 34\cdot 97 + 60\cdot 97^{2} + 67\cdot 97^{3} + 2\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 82\cdot 97 + 89\cdot 97^{2} + 44\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 40\cdot 97 + 44\cdot 97^{2} + 30\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 39\cdot 97 + 24\cdot 97^{2} + 23\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 55 + 42\cdot 97 + 38\cdot 97^{2} + 11\cdot 97^{3} + 86\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 59 + 74\cdot 97 + 29\cdot 97^{2} + 84\cdot 97^{3} + 38\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 77 + 91\cdot 97 + 62\cdot 97^{2} + 83\cdot 97^{3} + 96\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,8,7,2,6,3)$ |
| $(1,5,7,6)(2,3,4,8)$ |
| $(1,7)(2,4)(3,8)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $-1$ |
| $1$ | $4$ | $(1,5,7,6)(2,3,4,8)$ | $\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,6,7,5)(2,8,4,3)$ | $-\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,4,5,8,7,2,6,3)$ | $-\zeta_{8}$ |
| $1$ | $8$ | $(1,8,6,4,7,3,5,2)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,2,5,3,7,4,6,8)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,3,6,2,7,8,5,4)$ | $\zeta_{8}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
