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:
| $r_{ 1 }$ |
$=$ |
$ 17 + 9\cdot 89 + 79\cdot 89^{2} + 63\cdot 89^{3} + 75\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 a + 4 + \left(13 a + 11\right)\cdot 89 + 33 a\cdot 89^{2} + \left(9 a + 36\right)\cdot 89^{3} + \left(67 a + 29\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 43 a + 4 + \left(61 a + 87\right)\cdot 89 + \left(39 a + 36\right)\cdot 89^{2} + \left(68 a + 54\right)\cdot 89^{3} + \left(6 a + 18\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 46 a + 38 + \left(27 a + 29\right)\cdot 89 + \left(49 a + 75\right)\cdot 89^{2} + \left(20 a + 48\right)\cdot 89^{3} + \left(82 a + 86\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 a + 53 + \left(75 a + 6\right)\cdot 89 + \left(55 a + 41\right)\cdot 89^{2} + \left(79 a + 68\right)\cdot 89^{3} + \left(21 a + 44\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 63 + 34\cdot 89 + 34\cdot 89^{2} + 84\cdot 89^{3} + 11\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,6)$ |
| $(2,5)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(3,4)$ | $1$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $4$ | $3$ | $(1,3,2)(4,5,6)$ | $0$ |
| $4$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
| $4$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
