Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 12\cdot 97 + 16\cdot 97^{2} + 27\cdot 97^{3} + 61\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 18\cdot 97 + 62\cdot 97^{2} + 38\cdot 97^{3} + 36\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 58\cdot 97 + 97^{2} + 27\cdot 97^{3} + 42\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 55 + 5\cdot 97 + 83\cdot 97^{2} + 33\cdot 97^{3} + 42\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 14\cdot 97 + 32\cdot 97^{2} + 9\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 64 + 13\cdot 97 + 26\cdot 97^{2} + 46\cdot 97^{3} + 85\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 86 + 97 + 15\cdot 97^{2} + 12\cdot 97^{3} + 64\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 69\cdot 97 + 54\cdot 97^{2} + 96\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,8)(4,5)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,8)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,8,6,5)(2,3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
