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:
| $r_{ 1 }$ |
$=$ |
$ 29 a + 15 + \left(30 a + 24\right)\cdot 47 + \left(38 a + 19\right)\cdot 47^{2} + \left(46 a + 35\right)\cdot 47^{3} + \left(40 a + 30\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 a + 16 + \left(17 a + 27\right)\cdot 47 + \left(28 a + 19\right)\cdot 47^{2} + \left(28 a + 32\right)\cdot 47^{3} + \left(43 a + 36\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 a + 26 + \left(16 a + 9\right)\cdot 47 + \left(8 a + 19\right)\cdot 47^{2} + 43\cdot 47^{3} + \left(6 a + 18\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 34 + \left(29 a + 5\right)\cdot 47 + \left(18 a + 12\right)\cdot 47^{2} + \left(18 a + 14\right)\cdot 47^{3} + \left(3 a + 1\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 13\cdot 47 + 8\cdot 47^{2} + 15\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 + 13\cdot 47 + 15\cdot 47^{2} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,6)(3,5)$ |
| $(1,5,3)$ |
| $(3,5)(4,6)$ |
| $(1,6,5,4,3,2)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $3$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$0$ |
| $9$ |
$2$ |
$(2,6)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,5,3)(2,6,4)$ |
$-2$ |
| $2$ |
$3$ |
$(1,3,5)(2,6,4)$ |
$-2$ |
| $4$ |
$3$ |
$(1,5,3)$ |
$1$ |
| $6$ |
$6$ |
$(1,6,5,4,3,2)$ |
$0$ |
| $6$ |
$6$ |
$(1,4,3,2,5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
