Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 13 + \left(10 a + 10\right)\cdot 47 + \left(26 a + 6\right)\cdot 47^{2} + \left(36 a + 7\right)\cdot 47^{3} + \left(43 a + 1\right)\cdot 47^{4} + \left(21 a + 43\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 45 a + 33 + \left(22 a + 24\right)\cdot 47 + \left(28 a + 2\right)\cdot 47^{2} + \left(8 a + 10\right)\cdot 47^{3} + \left(10 a + 31\right)\cdot 47^{4} + \left(36 a + 10\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 15\cdot 47 + 39\cdot 47^{2} + 32\cdot 47^{3} + 40\cdot 47^{4} + 7\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 2 a + 29 + \left(24 a + 25\right)\cdot 47 + \left(18 a + 36\right)\cdot 47^{2} + \left(38 a + 45\right)\cdot 47^{3} + \left(36 a + 42\right)\cdot 47^{4} + \left(10 a + 25\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 31 + \left(36 a + 21\right)\cdot 47 + \left(20 a + 1\right)\cdot 47^{2} + \left(10 a + 7\right)\cdot 47^{3} + \left(3 a + 5\right)\cdot 47^{4} + \left(25 a + 43\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 43\cdot 47 + 7\cdot 47^{2} + 38\cdot 47^{3} + 19\cdot 47^{4} + 10\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,5)(2,6,4)$ |
| $(3,5)(4,6)$ |
| $(1,2)(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$-2$ |
| $3$ |
$2$ |
$(3,5)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,5)(2,6,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,6,5,2,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
