Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 26\cdot 47 + 26\cdot 47^{2} + 4\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 8 + \left(18 a + 31\right)\cdot 47 + \left(39 a + 3\right)\cdot 47^{2} + \left(11 a + 29\right)\cdot 47^{3} + \left(31 a + 24\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 36\cdot 47 + 43\cdot 47^{2} + 30\cdot 47^{3} + 7\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a + 13 + \left(21 a + 43\right)\cdot 47 + \left(13 a + 45\right)\cdot 47^{2} + \left(17 a + 20\right)\cdot 47^{3} + \left(34 a + 17\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 a + 25 + \left(28 a + 36\right)\cdot 47 + \left(7 a + 16\right)\cdot 47^{2} + \left(35 a + 13\right)\cdot 47^{3} + \left(15 a + 28\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 16 + \left(25 a + 14\right)\cdot 47 + \left(33 a + 4\right)\cdot 47^{2} + \left(29 a + 42\right)\cdot 47^{3} + \left(12 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(3,4)$ |
| $(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $-2$ |
| $6$ | $2$ | $(2,5)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $4$ | $3$ | $(1,2,5)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,4,2,6,5,3)$ | $1$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
