Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 4\cdot 47 + 14\cdot 47^{2} + 39\cdot 47^{3} + 43\cdot 47^{4} + 42\cdot 47^{5} + 15\cdot 47^{6} + 15\cdot 47^{7} + 6\cdot 47^{8} + 37\cdot 47^{9} + 33\cdot 47^{10} + 30\cdot 47^{11} + 43\cdot 47^{12} + 5\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 10 + \left(11 a + 44\right)\cdot 47 + \left(33 a + 6\right)\cdot 47^{2} + \left(35 a + 27\right)\cdot 47^{3} + \left(16 a + 20\right)\cdot 47^{4} + \left(12 a + 25\right)\cdot 47^{5} + \left(22 a + 31\right)\cdot 47^{6} + \left(22 a + 26\right)\cdot 47^{7} + \left(6 a + 16\right)\cdot 47^{8} + \left(2 a + 12\right)\cdot 47^{9} + \left(9 a + 29\right)\cdot 47^{10} + 34 a\cdot 47^{11} + \left(34 a + 17\right)\cdot 47^{12} + \left(5 a + 45\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 31 + \left(11 a + 30\right)\cdot 47 + \left(33 a + 31\right)\cdot 47^{2} + \left(35 a + 28\right)\cdot 47^{3} + \left(16 a + 28\right)\cdot 47^{4} + \left(12 a + 13\right)\cdot 47^{5} + \left(22 a + 30\right)\cdot 47^{6} + \left(22 a + 44\right)\cdot 47^{7} + \left(6 a + 39\right)\cdot 47^{8} + \left(2 a + 36\right)\cdot 47^{9} + \left(9 a + 1\right)\cdot 47^{10} + \left(34 a + 34\right)\cdot 47^{11} + \left(34 a + 41\right)\cdot 47^{12} + \left(5 a + 24\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 42\cdot 47 + 32\cdot 47^{2} + 7\cdot 47^{3} + 3\cdot 47^{4} + 4\cdot 47^{5} + 31\cdot 47^{6} + 31\cdot 47^{7} + 40\cdot 47^{8} + 9\cdot 47^{9} + 13\cdot 47^{10} + 16\cdot 47^{11} + 3\cdot 47^{12} + 41\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 44 a + 37 + \left(35 a + 2\right)\cdot 47 + \left(13 a + 40\right)\cdot 47^{2} + \left(11 a + 19\right)\cdot 47^{3} + \left(30 a + 26\right)\cdot 47^{4} + \left(34 a + 21\right)\cdot 47^{5} + \left(24 a + 15\right)\cdot 47^{6} + \left(24 a + 20\right)\cdot 47^{7} + \left(40 a + 30\right)\cdot 47^{8} + \left(44 a + 34\right)\cdot 47^{9} + \left(37 a + 17\right)\cdot 47^{10} + \left(12 a + 46\right)\cdot 47^{11} + \left(12 a + 29\right)\cdot 47^{12} + \left(41 a + 1\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 44 a + 16 + \left(35 a + 16\right)\cdot 47 + \left(13 a + 15\right)\cdot 47^{2} + \left(11 a + 18\right)\cdot 47^{3} + \left(30 a + 18\right)\cdot 47^{4} + \left(34 a + 33\right)\cdot 47^{5} + \left(24 a + 16\right)\cdot 47^{6} + \left(24 a + 2\right)\cdot 47^{7} + \left(40 a + 7\right)\cdot 47^{8} + \left(44 a + 10\right)\cdot 47^{9} + \left(37 a + 45\right)\cdot 47^{10} + \left(12 a + 12\right)\cdot 47^{11} + \left(12 a + 5\right)\cdot 47^{12} + \left(41 a + 22\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(5,6)$ |
| $(1,3,2)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(3,6)$ |
$1$ |
| $3$ |
$2$ |
$(2,5)(3,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(4,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)(3,6)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(2,3,5,6)$ |
$-1$ |
| $6$ |
$4$ |
$(1,5,4,2)(3,6)$ |
$1$ |
| $8$ |
$6$ |
$(1,3,5,4,6,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
