Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 20\cdot 97 + 46\cdot 97^{2} + 61\cdot 97^{3} + 78\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 17 + 80\cdot 97 + 62\cdot 97^{2} + 3\cdot 97^{3} + 8\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 23 + 95\cdot 97 + 79\cdot 97^{2} + 36\cdot 97^{3} + 35\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 84\cdot 97 + 32\cdot 97^{2} + 12\cdot 97^{3} + 46\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 57 + 22\cdot 97 + 62\cdot 97^{2} + 33\cdot 97^{3} + 88\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 61 + 7\cdot 97 + 12\cdot 97^{2} + 36\cdot 97^{3} + 26\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 80 + 66\cdot 97 + 52\cdot 97^{2} + 86\cdot 97^{3} + 77\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 94 + 10\cdot 97 + 39\cdot 97^{2} + 20\cdot 97^{3} + 27\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,8)(3,6)(5,7)$ |
| $(1,4)(3,6)$ |
| $(1,7,4,5)(2,3,8,6)$ |
| $(1,6,4,3)(2,5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,6,4,3)(2,5,8,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,6)(2,7,8,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,4,5)(2,3,8,6)$ | $0$ |
| $2$ | $4$ | $(1,3,4,6)(2,5,8,7)$ | $0$ |
| $2$ | $4$ | $(1,8,4,2)(3,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
