Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 18 + 27\cdot 97 + 50\cdot 97^{2} + 11\cdot 97^{3} + 17\cdot 97^{4} + 96\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 23\cdot 97 + 68\cdot 97^{2} + 97^{3} + 57\cdot 97^{4} + 80\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 50 + 25\cdot 97 + 18\cdot 97^{2} + 93\cdot 97^{3} + 91\cdot 97^{4} + 26\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 71 + 33\cdot 97 + 89\cdot 97^{2} + 5\cdot 97^{3} + 56\cdot 97^{4} + 21\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 74 + 13\cdot 97 + 85\cdot 97^{2} + 4\cdot 97^{3} + 97^{4} + 63\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 82 + 36\cdot 97 + 48\cdot 97^{2} + 97^{3} + 67\cdot 97^{4} + 88\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 84 + 32\cdot 97 + 87\cdot 97^{2} + 78\cdot 97^{3} + 21\cdot 97^{4} + 51\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 38\cdot 97^{2} + 93\cdot 97^{3} + 75\cdot 97^{4} + 56\cdot 97^{5} +O\left(97^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,4)(5,6,7,8)$ |
| $(3,8)(4,6)$ |
| $(1,2)(3,4)(5,7)(6,8)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
| $(1,7)(2,5)(3,8)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $-4$ |
| $2$ | $2$ | $(1,2)(3,4)(5,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(3,8)(4,6)$ | $0$ |
| $2$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $4$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $0$ |
| $4$ | $4$ | $(1,3,2,4)(5,6,7,8)$ | $0$ |
| $4$ | $4$ | $(1,4,2,3)(5,8,7,6)$ | $0$ |
| $4$ | $4$ | $(2,5)(3,4,8,6)$ | $0$ |
| $4$ | $4$ | $(2,5)(3,6,8,4)$ | $0$ |
| $4$ | $4$ | $(1,4,7,6)(2,8,5,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
