Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 41\cdot 97 + 5\cdot 97^{2} + 60\cdot 97^{3} + 20\cdot 97^{4} + 70\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 16\cdot 97 + 96\cdot 97^{2} + 29\cdot 97^{3} + 25\cdot 97^{4} + 92\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 13\cdot 97 + 15\cdot 97^{2} + 53\cdot 97^{3} + 12\cdot 97^{4} + 4\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 79\cdot 97^{2} + 2\cdot 97^{3} + 89\cdot 97^{4} + 74\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 67 + 48\cdot 97 + 58\cdot 97^{2} + 61\cdot 97^{3} + 39\cdot 97^{4} + 54\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 71 + 19\cdot 97 + 33\cdot 97^{2} + 50\cdot 97^{3} + 93\cdot 97^{4} + 89\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 77 + 64\cdot 97 + 80\cdot 97^{2} + 77\cdot 97^{3} + 79\cdot 97^{4} + 45\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 + 86\cdot 97 + 19\cdot 97^{2} + 52\cdot 97^{3} + 27\cdot 97^{4} + 53\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,4,3,8,6,2,5)$ |
| $(1,8)(2,4)(3,5)(6,7)$ |
| $(1,2)(4,8)(6,7)$ |
| $(1,2,8,4)(3,7,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,8)(6,7)$ | $0$ |
| $4$ | $2$ | $(1,5)(2,7)(3,8)(4,6)$ | $0$ |
| $2$ | $4$ | $(1,4,8,2)(3,6,5,7)$ | $0$ |
| $2$ | $8$ | $(1,7,4,3,8,6,2,5)$ | $\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,7,8,5,4,6)$ | $-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
