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:
| $r_{ 1 }$ |
$=$ |
$ 37 a + 57 + \left(35 a + 85\right)\cdot 89 + \left(33 a + 53\right)\cdot 89^{2} + \left(57 a + 3\right)\cdot 89^{3} + \left(57 a + 48\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 59 a + 49 + \left(39 a + 46\right)\cdot 89 + \left(29 a + 32\right)\cdot 89^{2} + \left(74 a + 44\right)\cdot 89^{3} + \left(32 a + 78\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 17 + \left(49 a + 87\right)\cdot 89 + \left(59 a + 20\right)\cdot 89^{2} + \left(14 a + 1\right)\cdot 89^{3} + \left(56 a + 56\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 72 + 62\cdot 89 + 49\cdot 89^{2} + 69\cdot 89^{3} + 2\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 52 a + 49 + \left(53 a + 29\right)\cdot 89 + \left(55 a + 74\right)\cdot 89^{2} + \left(31 a + 15\right)\cdot 89^{3} + \left(31 a + 38\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 44\cdot 89 + 35\cdot 89^{2} + 43\cdot 89^{3} + 43\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(2,3)$ |
| $(2,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,3)$ | $0$ |
| $9$ | $2$ | $(1,4)(2,3)$ | $0$ |
| $4$ | $3$ | $(2,3,6)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2,4,3)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,4,3,5,6)$ | $-1$ |
| $12$ | $6$ | $(1,4,5)(2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
