Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 12.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{3} + 3 x + 42 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 43\cdot 47 + 32\cdot 47^{2} + 43\cdot 47^{3} + 47^{4} + 30\cdot 47^{5} + 27\cdot 47^{6} + 34\cdot 47^{7} + 39\cdot 47^{8} + 31\cdot 47^{9} + 47^{10} + 43\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 15\cdot 47 + 10\cdot 47^{2} + 46\cdot 47^{3} + 35\cdot 47^{4} + 4\cdot 47^{5} + 7\cdot 47^{6} + 8\cdot 47^{7} + 15\cdot 47^{8} + 29\cdot 47^{9} + 26\cdot 47^{10} + 33\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 a^{2} + 8 a + 26 + \left(23 a^{2} + 39 a + 16\right)\cdot 47 + \left(32 a^{2} + 46 a + 20\right)\cdot 47^{2} + \left(44 a^{2} + 16 a + 3\right)\cdot 47^{3} + \left(35 a^{2} + 43 a + 36\right)\cdot 47^{4} + \left(2 a^{2} + 40\right)\cdot 47^{5} + \left(a^{2} + 40 a + 33\right)\cdot 47^{6} + \left(34 a^{2} + 28 a + 5\right)\cdot 47^{7} + \left(14 a^{2} + 25 a + 17\right)\cdot 47^{8} + \left(15 a^{2} + 17 a + 12\right)\cdot 47^{9} + \left(23 a^{2} + 17 a + 23\right)\cdot 47^{10} + \left(46 a^{2} + 4 a + 45\right)\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 a^{2} + 39 a + 12 + \left(45 a^{2} + 5 a + 13\right)\cdot 47 + \left(2 a^{2} + 35 a + 8\right)\cdot 47^{2} + \left(a^{2} + 27 a + 10\right)\cdot 47^{3} + \left(38 a^{2} + 36 a + 40\right)\cdot 47^{4} + \left(43 a^{2} + 20 a + 28\right)\cdot 47^{5} + \left(23 a^{2} + 15 a + 32\right)\cdot 47^{6} + \left(38 a^{2} + 27 a + 14\right)\cdot 47^{7} + \left(8 a^{2} + 8 a + 5\right)\cdot 47^{8} + \left(6 a^{2} + 32 a + 41\right)\cdot 47^{9} + \left(44 a^{2} + 37 a + 17\right)\cdot 47^{10} + \left(22 a^{2} + 6 a + 45\right)\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 39 + 27\cdot 47 + 2\cdot 47^{2} + 40\cdot 47^{3} + 23\cdot 47^{4} + 17\cdot 47^{5} + 21\cdot 47^{6} + 8\cdot 47^{7} + 9\cdot 47^{8} + 28\cdot 47^{9} + 16\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a^{2} + 29 a + 41 + \left(12 a^{2} + 42\right)\cdot 47 + \left(37 a^{2} + 17 a + 9\right)\cdot 47^{2} + \left(40 a^{2} + 15 a + 30\right)\cdot 47^{3} + \left(36 a^{2} + 27 a + 10\right)\cdot 47^{4} + \left(40 a^{2} + 9 a + 13\right)\cdot 47^{5} + \left(36 a^{2} + 19 a + 23\right)\cdot 47^{6} + \left(25 a^{2} + 20 a + 18\right)\cdot 47^{7} + \left(12 a^{2} + 10 a\right)\cdot 47^{8} + \left(44 a^{2} + 20 a + 30\right)\cdot 47^{9} + \left(10 a^{2} + 22 a + 35\right)\cdot 47^{10} + \left(36 a^{2} + 16 a + 10\right)\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 44 a^{2} + 5 a + 6 + \left(38 a^{2} + 25 a + 1\right)\cdot 47 + \left(4 a^{2} + 24 a + 39\right)\cdot 47^{2} + \left(6 a^{2} + 33 a + 7\right)\cdot 47^{3} + \left(3 a^{2} + 44 a + 37\right)\cdot 47^{4} + \left(5 a^{2} + 44 a + 35\right)\cdot 47^{5} + \left(6 a^{2} + 8 a + 8\right)\cdot 47^{6} + \left(40 a^{2} + 42 a\right)\cdot 47^{7} + \left(14 a^{2} + 42 a + 5\right)\cdot 47^{8} + \left(18 a^{2} + 34 a + 25\right)\cdot 47^{9} + \left(41 a^{2} + 26 a + 2\right)\cdot 47^{10} + \left(16 a^{2} + 35 a + 19\right)\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 14 a^{2} + 14 + \left(25 a^{2} + 2 a + 20\right)\cdot 47 + \left(11 a^{2} + 12 a + 25\right)\cdot 47^{2} + \left(a^{2} + 2 a + 10\right)\cdot 47^{3} + \left(20 a^{2} + 14 a + 4\right)\cdot 47^{4} + \left(25 a + 36\right)\cdot 47^{5} + \left(22 a^{2} + 38 a + 28\right)\cdot 47^{6} + \left(21 a^{2} + 37 a + 27\right)\cdot 47^{7} + \left(23 a^{2} + 12 a + 34\right)\cdot 47^{8} + \left(25 a^{2} + 44 a + 32\right)\cdot 47^{9} + \left(26 a^{2} + 38 a + 29\right)\cdot 47^{10} + \left(24 a^{2} + 35 a + 1\right)\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 12 a^{2} + 13 a + 36 + \left(42 a^{2} + 21 a + 7\right)\cdot 47 + \left(4 a^{2} + 5 a + 39\right)\cdot 47^{2} + \left(45 a + 42\right)\cdot 47^{3} + \left(7 a^{2} + 21 a + 44\right)\cdot 47^{4} + \left(a^{2} + 39 a + 27\right)\cdot 47^{5} + \left(4 a^{2} + 18 a + 4\right)\cdot 47^{6} + \left(28 a^{2} + 31 a + 23\right)\cdot 47^{7} + \left(19 a^{2} + 40 a + 14\right)\cdot 47^{8} + \left(31 a^{2} + 38 a + 4\right)\cdot 47^{9} + \left(41 a^{2} + 44 a + 3\right)\cdot 47^{10} + \left(40 a^{2} + 41 a + 20\right)\cdot 47^{11} +O\left(47^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,5)(3,4,8)(6,9,7)$ |
| $(1,4,9)(2,8,7)(3,6,5)$ |
| $(3,9)(4,7)(6,8)$ |
| $(3,8,4)(6,9,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(1,6)(2,9)(5,7)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,5)(3,4,8)(6,9,7)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,5,2)(3,8,4)(6,7,9)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $6$ |
$3$ |
$(1,4,9)(2,8,7)(3,6,5)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,3,9)(2,4,7)(5,8,6)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(3,8,4)(6,9,7)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,9,8)(2,7,3)(4,5,6)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,7,2,6,5,9)(3,8,4)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $9$ |
$6$ |
$(1,9,5,6,2,7)(3,4,8)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
