Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 74\cdot 97 + 85\cdot 97^{2} + 65\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 76\cdot 97 + 75\cdot 97^{2} + 31\cdot 97^{3} + 20\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 82\cdot 97 + 86\cdot 97^{2} + 59\cdot 97^{3} + 20\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 + 28\cdot 97 + 34\cdot 97^{2} + 84\cdot 97^{3} + 90\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 + 64\cdot 97 + 88\cdot 97^{2} + 28\cdot 97^{3} + 49\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 65 + 45\cdot 97 + 6\cdot 97^{2} + 66\cdot 97^{3} + 76\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 77 + 38\cdot 97 + 22\cdot 97^{2} + 97^{3} + 72\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 89 + 74\cdot 97 + 84\cdot 97^{2} + 17\cdot 97^{3} + 90\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3,2,7)(4,8,5,6)$ |
| $(1,4)(2,5)(3,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,7)(4,5)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,5)(3,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,8)(3,5)(4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,7)(4,8,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
