Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 58\cdot 103 + 4\cdot 103^{2} + 68\cdot 103^{3} + 35\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 27 + 66\cdot 103 + 34\cdot 103^{2} + 18\cdot 103^{3} + 84\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 59\cdot 103 + 71\cdot 103^{2} + 32\cdot 103^{3} + 88\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 60 + 83\cdot 103 + 54\cdot 103^{2} + 56\cdot 103^{3} + 53\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 85 + 99\cdot 103 + 44\cdot 103^{2} + 98\cdot 103^{3} + 82\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 91 + 99\cdot 103 + 86\cdot 103^{2} + 79\cdot 103^{3} + 3\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 97 + 6\cdot 103 + 99\cdot 103^{2} + 66\cdot 103^{3} + 2\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 101 + 40\cdot 103 + 15\cdot 103^{2} + 94\cdot 103^{3} + 60\cdot 103^{4} +O\left(103^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(2,5)(3,4)$ |
| $(1,7,8,6)(2,4,5,3)$ |
| $(1,8)(2,5)(3,4)(6,7)$ |
| $(1,4)(2,6)(3,8)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(2,5)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,8)(3,7)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,7,8,6)(2,4,5,3)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,7)(2,3,5,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,4)(2,6,5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,8,2)(3,6,4,7)$ | $0$ |
| $2$ | $4$ | $(1,7,8,6)(2,3,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
