Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 + 77\cdot 103 + 101\cdot 103^{2} + 101\cdot 103^{3} + 43\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 + 61\cdot 103 + 58\cdot 103^{2} + 62\cdot 103^{3} + 47\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 + 55\cdot 103 + 102\cdot 103^{2} + 88\cdot 103^{3} + 87\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 92\cdot 103 + 56\cdot 103^{2} + 30\cdot 103^{3} + 34\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 69 + 8\cdot 103 + 52\cdot 103^{2} + 79\cdot 103^{3} + 36\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 89 + 83\cdot 103 + 47\cdot 103^{2} + 87\cdot 103^{3} + 39\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 98 + 11\cdot 103 + 46\cdot 103^{2} + 55\cdot 103^{3} + 26\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 99 + 20\cdot 103 + 49\cdot 103^{2} + 8\cdot 103^{3} + 95\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,7)(4,8,6,5)$ |
| $(1,4)(2,5)(3,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,7)(4,6)(5,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,5)(3,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,8)(4,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,7)(4,8,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
