Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 54\cdot 97 + 2\cdot 97^{2} + 89\cdot 97^{3} + 30\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 44\cdot 97 + 23\cdot 97^{2} + 87\cdot 97^{3} + 94\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 24 + 55\cdot 97 + 7\cdot 97^{2} + 42\cdot 97^{3} + 56\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 58\cdot 97 + 20\cdot 97^{2} + 25\cdot 97^{3} + 51\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 38\cdot 97 + 76\cdot 97^{2} + 71\cdot 97^{3} + 45\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 73 + 41\cdot 97 + 89\cdot 97^{2} + 54\cdot 97^{3} + 40\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 81 + 52\cdot 97 + 73\cdot 97^{2} + 9\cdot 97^{3} + 2\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 83 + 42\cdot 97 + 94\cdot 97^{2} + 7\cdot 97^{3} + 66\cdot 97^{4} +O\left(97^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,4)(2,7)(3,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,7)(3,6)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,5,6)(2,4,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
