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 a + 6 + \left(34 a + 22\right)\cdot 47 + \left(38 a + 34\right)\cdot 47^{2} + \left(21 a + 26\right)\cdot 47^{3} + \left(29 a + 30\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 42\cdot 47 + 28\cdot 47^{2} + 35\cdot 47^{3} + 42\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 39\cdot 47 + 36\cdot 47^{2} + 46\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 45 a + 36 + \left(45 a + 3\right)\cdot 47 + \left(16 a + 11\right)\cdot 47^{2} + \left(21 a + 34\right)\cdot 47^{3} + \left(43 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 2 a + 32 + \left(a + 3\right)\cdot 47 + \left(30 a + 46\right)\cdot 47^{2} + \left(25 a + 12\right)\cdot 47^{3} + \left(3 a + 16\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 33 a + 34 + \left(12 a + 29\right)\cdot 47 + \left(8 a + 30\right)\cdot 47^{2} + \left(25 a + 31\right)\cdot 47^{3} + \left(17 a + 20\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,6)$ |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(4,5)$ |
$0$ |
| $9$ |
$2$ |
$(2,6)(4,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,6)(3,4,5)$ |
$1$ |
| $4$ |
$3$ |
$(3,4,5)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,4,6,5)$ |
$0$ |
| $12$ |
$6$ |
$(1,3,2,4,6,5)$ |
$1$ |
| $12$ |
$6$ |
$(1,2,6)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
