Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 81\cdot 97 + 35\cdot 97^{2} + 85\cdot 97^{3} + 49\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 66\cdot 97 + 24\cdot 97^{2} + 15\cdot 97^{3} + 37\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 35\cdot 97 + 21\cdot 97^{2} + 41\cdot 97^{3} + 90\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 20\cdot 97 + 76\cdot 97^{2} + 65\cdot 97^{3} + 49\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 59\cdot 97 + 21\cdot 97^{2} + 3\cdot 97^{3} + 82\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 51 + 76\cdot 97 + 36\cdot 97^{2} + 57\cdot 97^{3} + 70\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 60\cdot 97 + 5\cdot 97^{2} + 96\cdot 97^{3} + 78\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 86 + 85\cdot 97 + 68\cdot 97^{2} + 23\cdot 97^{3} + 26\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,7)(3,6)(5,8)$ |
| $(1,2,5,3)(4,6,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,5,3)(4,6,8,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
