Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 69\cdot 97 + 92\cdot 97^{2} + 28\cdot 97^{3} + 16\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 56\cdot 97 + 43\cdot 97^{2} + 65\cdot 97^{3} + 8\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 84\cdot 97 + 89\cdot 97^{2} + 69\cdot 97^{3} + 28\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 43 + 49\cdot 97 + 7\cdot 97^{2} + 75\cdot 97^{3} + 62\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 54 + 47\cdot 97 + 89\cdot 97^{2} + 21\cdot 97^{3} + 34\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 12\cdot 97 + 7\cdot 97^{2} + 27\cdot 97^{3} + 68\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 79 + 40\cdot 97 + 53\cdot 97^{2} + 31\cdot 97^{3} + 88\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 90 + 27\cdot 97 + 4\cdot 97^{2} + 68\cdot 97^{3} + 80\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,4,3,7)(2,8,5,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,4)(3,6)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,4,3,7)(2,8,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
