Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 50 a + 3 + \left(71 a + 86\right)\cdot 89 + \left(23 a + 85\right)\cdot 89^{2} + \left(51 a + 54\right)\cdot 89^{3} + \left(8 a + 84\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 41 + 55\cdot 89 + 66\cdot 89^{2} + 14\cdot 89^{3} + 85\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 13 + \left(47 a + 62\right)\cdot 89 + \left(57 a + 44\right)\cdot 89^{2} + \left(80 a + 13\right)\cdot 89^{3} + \left(77 a + 79\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a + 86 + \left(17 a + 2\right)\cdot 89 + \left(65 a + 3\right)\cdot 89^{2} + \left(37 a + 34\right)\cdot 89^{3} + \left(80 a + 4\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 48 + 33\cdot 89 + 22\cdot 89^{2} + 74\cdot 89^{3} + 3\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 80 a + 76 + \left(41 a + 26\right)\cdot 89 + \left(31 a + 44\right)\cdot 89^{2} + \left(8 a + 75\right)\cdot 89^{3} + \left(11 a + 9\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,6,5)(2,4,3)$ |
| $(1,2,4,5)$ |
| $(1,2,3)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,5)(4,6)$ | $1$ |
| $8$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $6$ | $4$ | $(1,2,4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
