Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 89 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 89 }$: $ x^{2} + 82 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 72 a + 46 + \left(65 a + 4\right)\cdot 89 + \left(75 a + 18\right)\cdot 89^{2} + \left(56 a + 45\right)\cdot 89^{3} + \left(22 a + 70\right)\cdot 89^{4} + \left(18 a + 14\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 26 a + 67 + \left(24 a + 10\right)\cdot 89 + \left(16 a + 2\right)\cdot 89^{2} + \left(54 a + 9\right)\cdot 89^{3} + \left(54 a + 69\right)\cdot 89^{4} + \left(45 a + 50\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 a + 71 + \left(64 a + 65\right)\cdot 89 + \left(72 a + 2\right)\cdot 89^{2} + \left(34 a + 16\right)\cdot 89^{3} + \left(34 a + 41\right)\cdot 89^{4} + \left(43 a + 48\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 + 49\cdot 89 + 70\cdot 89^{2} + 17\cdot 89^{3} + 41\cdot 89^{4} + 35\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 16 + \left(23 a + 37\right)\cdot 89 + \left(13 a + 37\right)\cdot 89^{2} + \left(32 a + 11\right)\cdot 89^{3} + \left(66 a + 83\right)\cdot 89^{4} + \left(70 a + 30\right)\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 + 10\cdot 89 + 47\cdot 89^{2} + 78\cdot 89^{3} + 50\cdot 89^{4} + 86\cdot 89^{5} +O\left(89^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,5,6)$ |
| $(3,4)(5,6)$ |
| $(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(1,2)$ | $1$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $-1$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(1,5,2,3)$ | $-1$ |
| $6$ | $4$ | $(1,6,2,4)(3,5)$ | $1$ |
| $8$ | $6$ | $(1,5,6,2,3,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
