Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 89\cdot 97 + 32\cdot 97^{2} + 67\cdot 97^{3} + 83\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 32\cdot 97 + 43\cdot 97^{2} + 82\cdot 97^{3} + 78\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 3\cdot 97 + 39\cdot 97^{2} + 31\cdot 97^{3} + 40\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 25\cdot 97 + 82\cdot 97^{2} + 6\cdot 97^{3} + 86\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 + 71\cdot 97 + 14\cdot 97^{2} + 90\cdot 97^{3} + 10\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 92 + 93\cdot 97 + 57\cdot 97^{2} + 65\cdot 97^{3} + 56\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 94 + 64\cdot 97 + 53\cdot 97^{2} + 14\cdot 97^{3} + 18\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 7\cdot 97 + 64\cdot 97^{2} + 29\cdot 97^{3} + 13\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(2,7)(4,5)$ |
| $(1,7)(2,8)(3,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
