Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 40 + 4\cdot 97 + 89\cdot 97^{2} + 85\cdot 97^{3} + 59\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 53 + 23\cdot 97 + 94\cdot 97^{2} + 7\cdot 97^{3} + 66\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 67 + 31\cdot 97 + 88\cdot 97^{2} + 16\cdot 97^{3} + 27\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 78 + 79\cdot 97 + 27\cdot 97^{2} + 71\cdot 97^{3} + 74\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 80 + 50\cdot 97 + 93\cdot 97^{2} + 35\cdot 97^{3} + 33\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 83 + 10\cdot 97 + 84\cdot 97^{2} + 46\cdot 97^{3} + 78\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 90 + 30\cdot 97 + 24\cdot 97^{2} + 25\cdot 97^{3} + 22\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 95 + 58\cdot 97 + 80\cdot 97^{2} + 26\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,5,8)(2,7,3,6)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,4,5,8)(2,7,3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
