Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 86\cdot 97 + 66\cdot 97^{2} + 47\cdot 97^{3} + 89\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 54 + 34\cdot 97 + 47\cdot 97^{2} + 19\cdot 97^{3} + 77\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 73 + 93\cdot 97 + 28\cdot 97^{2} + 45\cdot 97^{3} + 94\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 75 + 74\cdot 97 + 14\cdot 97^{2} + 27\cdot 97^{3} + 77\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 94\cdot 97 + 58\cdot 97^{2} + 31\cdot 97^{3} + 37\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 82 + 86\cdot 97 + 72\cdot 97^{2} + 18\cdot 97^{3} + 2\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 85 + 41\cdot 97 + 77\cdot 97^{2} + 2\cdot 97^{3} + 17\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 + 68\cdot 97 + 20\cdot 97^{2} + 97^{3} + 90\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,6)(3,7)(4,5)$ |
| $(1,3)(4,5)(7,8)$ |
| $(1,4,3,6,8,5,7,2)$ |
| $(1,7,8,3)(2,5,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,6)(3,7)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,3)(4,5)(7,8)$ | $0$ |
| $2$ | $4$ | $(1,3,8,7)(2,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,5,8,4)(2,3,6,7)$ | $0$ |
| $2$ | $8$ | $(1,4,3,6,8,5,7,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
