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:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 2 + \left(2 a + 28\right)\cdot 47 + \left(18 a + 15\right)\cdot 47^{2} + \left(13 a + 31\right)\cdot 47^{3} + \left(42 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 32 a + 21 + \left(10 a + 38\right)\cdot 47 + \left(a + 14\right)\cdot 47^{2} + \left(6 a + 42\right)\cdot 47^{3} + \left(30 a + 29\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 a + 38 + \left(36 a + 27\right)\cdot 47 + \left(45 a + 6\right)\cdot 47^{2} + \left(40 a + 6\right)\cdot 47^{3} + \left(16 a + 37\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 + 6\cdot 47 + 29\cdot 47^{2} + 22\cdot 47^{3} + 41\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 a + 42 + \left(44 a + 12\right)\cdot 47 + \left(28 a + 2\right)\cdot 47^{2} + \left(33 a + 40\right)\cdot 47^{3} + \left(4 a + 14\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 27\cdot 47 + 25\cdot 47^{2} + 45\cdot 47^{3} + 26\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,4)$ |
| $(1,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(3,6)$ | $0$ |
| $9$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $4$ | $3$ | $(1,4,5)$ | $-2$ |
| $4$ | $3$ | $(1,4,5)(2,3,6)$ | $1$ |
| $18$ | $4$ | $(1,2)(3,5,6,4)$ | $0$ |
| $12$ | $6$ | $(1,3,4,6,5,2)$ | $-1$ |
| $12$ | $6$ | $(1,4,5)(3,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
