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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 84\cdot 89 + 4\cdot 89^{2} + 56\cdot 89^{3} + 86\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 68 a + 36 + \left(34 a + 42\right)\cdot 89 + \left(79 a + 39\right)\cdot 89^{2} + \left(11 a + 5\right)\cdot 89^{3} + \left(52 a + 7\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 a + 18 + \left(16 a + 8\right)\cdot 89 + \left(66 a + 73\right)\cdot 89^{2} + \left(54 a + 60\right)\cdot 89^{3} + \left(47 a + 5\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 + 27\cdot 89 + 48\cdot 89^{2} + 24\cdot 89^{3} + 57\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 67 + \left(54 a + 39\right)\cdot 89 + \left(9 a + 26\right)\cdot 89^{2} + \left(77 a + 9\right)\cdot 89^{3} + \left(36 a + 4\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 a + 82 + \left(72 a + 64\right)\cdot 89 + \left(22 a + 74\right)\cdot 89^{2} + \left(34 a + 21\right)\cdot 89^{3} + \left(41 a + 17\right)\cdot 89^{4} +O\left(89^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,2,4,5,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $15$ | $2$ | $(1,4)(2,5)$ | $-2$ |
| $20$ | $3$ | $(1,5,2)(3,6,4)$ | $0$ |
| $30$ | $4$ | $(2,6,4,5)$ | $0$ |
| $24$ | $5$ | $(1,4,6,2,3)$ | $1$ |
| $20$ | $6$ | $(1,6,5,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
