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