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