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:
| $r_{ 1 }$ |
$=$ |
$ 26 a + 35 + 47 + \left(31 a + 26\right)\cdot 47^{2} + \left(25 a + 37\right)\cdot 47^{3} + \left(24 a + 26\right)\cdot 47^{4} + \left(19 a + 14\right)\cdot 47^{5} + \left(15 a + 39\right)\cdot 47^{6} + \left(25 a + 27\right)\cdot 47^{7} + \left(19 a + 41\right)\cdot 47^{8} + \left(18 a + 39\right)\cdot 47^{9} + \left(8 a + 34\right)\cdot 47^{10} + \left(30 a + 10\right)\cdot 47^{11} + \left(5 a + 37\right)\cdot 47^{12} + \left(23 a + 15\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 38 + 42\cdot 47 + 21\cdot 47^{2} + 15\cdot 47^{3} + 27\cdot 47^{4} + 44\cdot 47^{5} + 47^{6} + 47^{7} + 6\cdot 47^{8} + 3\cdot 47^{9} + 21\cdot 47^{10} + 46\cdot 47^{11} + 14\cdot 47^{12} + 26\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 4\cdot 47 + 25\cdot 47^{2} + 31\cdot 47^{3} + 19\cdot 47^{4} + 2\cdot 47^{5} + 45\cdot 47^{6} + 45\cdot 47^{7} + 40\cdot 47^{8} + 43\cdot 47^{9} + 25\cdot 47^{10} + 32\cdot 47^{12} + 20\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 21 a + 40 + \left(46 a + 23\right)\cdot 47 + \left(15 a + 40\right)\cdot 47^{2} + \left(21 a + 10\right)\cdot 47^{3} + \left(22 a + 3\right)\cdot 47^{4} + \left(27 a + 29\right)\cdot 47^{5} + \left(31 a + 3\right)\cdot 47^{6} + \left(21 a + 16\right)\cdot 47^{7} + \left(27 a + 8\right)\cdot 47^{8} + \left(28 a + 10\right)\cdot 47^{9} + \left(38 a + 33\right)\cdot 47^{10} + \left(16 a + 15\right)\cdot 47^{11} + \left(41 a + 18\right)\cdot 47^{12} + \left(23 a + 9\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 13 + \left(46 a + 45\right)\cdot 47 + \left(15 a + 20\right)\cdot 47^{2} + \left(21 a + 9\right)\cdot 47^{3} + \left(22 a + 20\right)\cdot 47^{4} + \left(27 a + 32\right)\cdot 47^{5} + \left(31 a + 7\right)\cdot 47^{6} + \left(21 a + 19\right)\cdot 47^{7} + \left(27 a + 5\right)\cdot 47^{8} + \left(28 a + 7\right)\cdot 47^{9} + \left(38 a + 12\right)\cdot 47^{10} + \left(16 a + 36\right)\cdot 47^{11} + \left(41 a + 9\right)\cdot 47^{12} + \left(23 a + 31\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 26 a + 8 + 23\cdot 47 + \left(31 a + 6\right)\cdot 47^{2} + \left(25 a + 36\right)\cdot 47^{3} + \left(24 a + 43\right)\cdot 47^{4} + \left(19 a + 17\right)\cdot 47^{5} + \left(15 a + 43\right)\cdot 47^{6} + \left(25 a + 30\right)\cdot 47^{7} + \left(19 a + 38\right)\cdot 47^{8} + \left(18 a + 36\right)\cdot 47^{9} + \left(8 a + 13\right)\cdot 47^{10} + \left(30 a + 31\right)\cdot 47^{11} + \left(5 a + 28\right)\cdot 47^{12} + \left(23 a + 37\right)\cdot 47^{13} +O\left(47^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(1,2)(3,5)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,3)(4,6)$ |
$-3$ |
| $3$ |
$2$ |
$(2,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(2,3)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,5,3)$ |
$1$ |
| $6$ |
$4$ |
$(1,6,5,4)(2,3)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,2,5,6,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
