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:
| $r_{ 1 }$ |
$=$ |
$ a + 25 + \left(55 a + 39\right)\cdot 89 + \left(67 a + 78\right)\cdot 89^{2} + \left(3 a + 72\right)\cdot 89^{3} + 38\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 88 a + 32 + \left(33 a + 67\right)\cdot 89 + \left(21 a + 51\right)\cdot 89^{2} + \left(85 a + 31\right)\cdot 89^{3} + \left(88 a + 35\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 a + 81 + \left(56 a + 49\right)\cdot 89 + \left(82 a + 17\right)\cdot 89^{2} + \left(73 a + 12\right)\cdot 89^{3} + \left(20 a + 61\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 50 a + 43 + \left(13 a + 88\right)\cdot 89 + \left(84 a + 35\right)\cdot 89^{2} + \left(23 a + 32\right)\cdot 89^{3} + \left(65 a + 5\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 68 a + 50 + \left(32 a + 66\right)\cdot 89 + \left(6 a + 5\right)\cdot 89^{2} + \left(15 a + 2\right)\cdot 89^{3} + \left(68 a + 44\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 37 + \left(75 a + 44\right)\cdot 89 + \left(4 a + 77\right)\cdot 89^{2} + \left(65 a + 26\right)\cdot 89^{3} + \left(23 a + 82\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(1,3,6,2,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,6,2,5,4)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,4,5,2,6,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
