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