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:
| $r_{ 1 }$ |
$=$ |
$ 5 a + 40 + \left(23 a + 21\right)\cdot 47 + \left(14 a + 9\right)\cdot 47^{2} + \left(10 a + 21\right)\cdot 47^{3} + \left(3 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 3 + \left(23 a + 16\right)\cdot 47 + \left(32 a + 15\right)\cdot 47^{2} + \left(36 a + 27\right)\cdot 47^{3} + \left(43 a + 16\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 20\cdot 47 + 15\cdot 47^{2} + 39\cdot 47^{3} + 43\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 44 a + 25 + \left(3 a + 31\right)\cdot 47 + \left(38 a + 26\right)\cdot 47^{2} + \left(11 a + 34\right)\cdot 47^{3} + \left(25 a + 5\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 + 9\cdot 47 + 22\cdot 47^{2} + 45\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 19 + \left(43 a + 42\right)\cdot 47 + \left(8 a + 4\right)\cdot 47^{2} + \left(35 a + 20\right)\cdot 47^{3} + \left(21 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(2,5)$ | $-2$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $-2$ |
| $4$ | $3$ | $(3,4,6)$ | $1$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,5,6)$ | $0$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
