Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 13\cdot 97 + 50\cdot 97^{2} + 89\cdot 97^{3} + 64\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 20\cdot 97 + 47\cdot 97^{2} + 68\cdot 97^{3} + 67\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 50\cdot 97 + 55\cdot 97^{2} + 32\cdot 97^{3} + 48\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 57\cdot 97 + 52\cdot 97^{2} + 11\cdot 97^{3} + 51\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 + 80\cdot 97 + 16\cdot 97^{2} + 72\cdot 97^{3} + 57\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 69 + 5\cdot 97 + 69\cdot 97^{2} + 77\cdot 97^{3} + 36\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 80 + 20\cdot 97 + 22\cdot 97^{2} + 15\cdot 97^{3} + 41\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 86 + 42\cdot 97 + 74\cdot 97^{2} + 20\cdot 97^{3} + 20\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,8)(4,6)(5,7)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,8)(4,6)(5,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,5,4)(2,3,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
