Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 63\cdot 97 + 59\cdot 97^{2} + 59\cdot 97^{3} + 29\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 36\cdot 97 + 80\cdot 97^{2} + 86\cdot 97^{3} + 73\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 82\cdot 97 + 17\cdot 97^{2} + 38\cdot 97^{3} + 91\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 + 26\cdot 97 + 41\cdot 97^{2} + 33\cdot 97^{3} + 10\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 63 + 28\cdot 97 + 44\cdot 97^{2} + 88\cdot 97^{3} + 82\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 68 + 5\cdot 97 + 28\cdot 97^{2} + 82\cdot 97^{3} + 26\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 69 + 75\cdot 97 + 48\cdot 97^{2} + 12\cdot 97^{3} + 71\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 79 + 69\cdot 97 + 67\cdot 97^{2} + 83\cdot 97^{3} + 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,8)(2,4,6,5)$ |
| $(1,2)(3,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,6)(3,8)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,3)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,8)(2,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
