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:
| $r_{ 1 }$ |
$=$ |
$ 54 a + 81 + \left(10 a + 84\right)\cdot 89 + \left(25 a + 80\right)\cdot 89^{2} + \left(31 a + 68\right)\cdot 89^{3} + \left(53 a + 51\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 59 + 53\cdot 89 + 78\cdot 89^{2} + 63\cdot 89^{3} + 62\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 83 + 27\cdot 89 + 81\cdot 89^{2} + 33\cdot 89^{3} + 26\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 64 a + 59 + \left(62 a + 32\right)\cdot 89 + \left(34 a + 22\right)\cdot 89^{2} + \left(45 a + 44\right)\cdot 89^{3} + \left(80 a + 7\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a + 14 + \left(78 a + 16\right)\cdot 89 + \left(63 a + 68\right)\cdot 89^{2} + \left(57 a + 84\right)\cdot 89^{3} + \left(35 a + 37\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 62 + \left(26 a + 51\right)\cdot 89 + \left(54 a + 24\right)\cdot 89^{2} + \left(43 a + 60\right)\cdot 89^{3} + \left(8 a + 80\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,5)$ | $-2$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)$ | $1$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $0$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
