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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 39 + \left(20 a + 66\right)\cdot 89 + \left(17 a + 22\right)\cdot 89^{2} + \left(86 a + 81\right)\cdot 89^{3} + \left(8 a + 3\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 40 a + 62 + \left(56 a + 66\right)\cdot 89 + \left(14 a + 74\right)\cdot 89^{2} + \left(22 a + 85\right)\cdot 89^{3} + \left(6 a + 48\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 49 a + 75 + \left(32 a + 65\right)\cdot 89 + \left(74 a + 31\right)\cdot 89^{2} + \left(66 a + 48\right)\cdot 89^{3} + \left(82 a + 70\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 51 a + 38 + \left(68 a + 82\right)\cdot 89 + \left(71 a + 33\right)\cdot 89^{2} + \left(2 a + 44\right)\cdot 89^{3} + \left(80 a + 69\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 78 a + 65 + \left(76 a + 29\right)\cdot 89 + \left(31 a + 23\right)\cdot 89^{2} + \left(19 a + 85\right)\cdot 89^{3} + \left(15 a + 37\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 77 + \left(12 a + 44\right)\cdot 89 + \left(57 a + 80\right)\cdot 89^{2} + \left(69 a + 10\right)\cdot 89^{3} + \left(73 a + 36\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)(3,5,4)$ |
| $(1,4)(2,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,6,4,2,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,2,4,6,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
