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