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:
| $r_{ 1 }$ |
$=$ |
$ 15 a^{2} + 17 a + 14 + \left(17 a^{2} + 18 a + 15\right)\cdot 31 + \left(2 a^{2} + 9 a + 13\right)\cdot 31^{2} + \left(10 a^{2} + 15 a + 19\right)\cdot 31^{3} + \left(13 a^{2} + 19 a + 29\right)\cdot 31^{4} + \left(23 a + 16\right)\cdot 31^{5} + \left(3 a^{2} + 18 a + 28\right)\cdot 31^{6} + \left(13 a^{2} + 4 a + 15\right)\cdot 31^{7} + \left(26 a^{2} + 12 a + 8\right)\cdot 31^{8} + \left(6 a^{2} + 14 a + 5\right)\cdot 31^{9} + \left(11 a^{2} + 5 a + 14\right)\cdot 31^{10} + \left(16 a^{2} + 19 a + 29\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a^{2} + 26 a + 7 + \left(a^{2} + 25 a + 15\right)\cdot 31 + \left(18 a^{2} + 8 a + 13\right)\cdot 31^{2} + \left(17 a^{2} + 24\right)\cdot 31^{3} + \left(12 a^{2} + 28 a + 18\right)\cdot 31^{4} + \left(13 a^{2} + 8 a + 25\right)\cdot 31^{5} + \left(4 a^{2} + 18 a + 8\right)\cdot 31^{6} + \left(24 a^{2} + 10 a + 23\right)\cdot 31^{7} + \left(4 a^{2} + 15 a + 14\right)\cdot 31^{8} + \left(4 a^{2} + 6 a + 3\right)\cdot 31^{9} + \left(18 a^{2} + a + 29\right)\cdot 31^{10} + \left(29 a^{2} + 8 a + 27\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a^{2} + 19 a + 22 + \left(11 a^{2} + 17 a + 11\right)\cdot 31 + \left(10 a^{2} + 12 a + 8\right)\cdot 31^{2} + \left(3 a^{2} + 15 a + 25\right)\cdot 31^{3} + \left(5 a^{2} + 14 a + 13\right)\cdot 31^{4} + \left(17 a^{2} + 29 a + 7\right)\cdot 31^{5} + \left(23 a^{2} + 24 a + 11\right)\cdot 31^{6} + \left(24 a^{2} + 15 a + 13\right)\cdot 31^{7} + \left(30 a^{2} + 3 a + 11\right)\cdot 31^{8} + \left(19 a^{2} + 10 a + 24\right)\cdot 31^{9} + \left(a^{2} + 24 a + 7\right)\cdot 31^{10} + \left(16 a^{2} + 3 a + 29\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a^{2} + 23 a + 11 + \left(25 a^{2} + 28 a + 6\right)\cdot 31 + \left(17 a^{2} + 12 a + 7\right)\cdot 31^{2} + \left(12 a^{2} + 14 a + 19\right)\cdot 31^{3} + \left(5 a^{2} + 10 a + 28\right)\cdot 31^{4} + \left(18 a^{2} + 20 a + 9\right)\cdot 31^{5} + \left(19 a^{2} + a + 6\right)\cdot 31^{6} + \left(9 a^{2} + 10 a + 21\right)\cdot 31^{7} + \left(14 a^{2} + 23 a + 20\right)\cdot 31^{8} + \left(30 a^{2} + 22 a + 1\right)\cdot 31^{9} + \left(17 a^{2} + 24 a + 27\right)\cdot 31^{10} + \left(12 a^{2} + 9 a + 13\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 18\cdot 31 + 29\cdot 31^{2} + 25\cdot 31^{3} + 3\cdot 31^{4} + 12\cdot 31^{5} + 17\cdot 31^{6} + 20\cdot 31^{7} + 26\cdot 31^{8} + 3\cdot 31^{9} + 16\cdot 31^{10} + 25\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 5 a^{2} + 29 a + 29 + \left(2 a^{2} + 28 a + 21\right)\cdot 31 + \left(22 a^{2} + 26 a + 30\right)\cdot 31^{2} + \left(16 a^{2} + 19 a + 21\right)\cdot 31^{3} + \left(24 a^{2} + 17 a + 20\right)\cdot 31^{4} + \left(15 a^{2} + 25 a + 18\right)\cdot 31^{5} + \left(4 a^{2} + 17 a + 6\right)\cdot 31^{6} + \left(27 a^{2} + 13 a + 12\right)\cdot 31^{7} + \left(5 a^{2} + 28 a + 25\right)\cdot 31^{8} + \left(14 a^{2} + 8 a + 21\right)\cdot 31^{9} + \left(21 a^{2} + 15 a + 8\right)\cdot 31^{10} + \left(27 a^{2} + 15 a + 3\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 14 + 16\cdot 31 + 14\cdot 31^{2} + 13\cdot 31^{3} + 25\cdot 31^{4} + 29\cdot 31^{5} + 24\cdot 31^{6} + 22\cdot 31^{7} + 4\cdot 31^{8} + 17\cdot 31^{9} + 14\cdot 31^{10} + 5\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 17 a^{2} + 10 a + 6 + \left(3 a^{2} + 4 a + 2\right)\cdot 31 + \left(22 a^{2} + 22 a + 10\right)\cdot 31^{2} + \left(a^{2} + 27 a + 22\right)\cdot 31^{3} + \left(a^{2} + 2 a + 25\right)\cdot 31^{4} + \left(28 a^{2} + 16 a + 26\right)\cdot 31^{5} + \left(6 a^{2} + 11 a + 28\right)\cdot 31^{6} + \left(25 a^{2} + 7 a + 10\right)\cdot 31^{7} + \left(10 a^{2} + 10 a + 18\right)\cdot 31^{8} + \left(17 a^{2} + 30 a + 13\right)\cdot 31^{9} + \left(22 a^{2} + 21 a + 9\right)\cdot 31^{10} + \left(21 a^{2} + 5 a + 30\right)\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 2 + 16\cdot 31 + 27\cdot 31^{2} + 13\cdot 31^{3} + 19\cdot 31^{4} + 7\cdot 31^{5} + 22\cdot 31^{6} + 14\cdot 31^{7} + 24\cdot 31^{8} + 31^{9} + 28\cdot 31^{10} + 20\cdot 31^{11} +O\left(31^{ 12 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,6,8)(5,9,7)$ |
| $(4,9)(5,8)(6,7)$ |
| $(4,8,6)(5,9,7)$ |
| $(1,6,9)(2,8,7)(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $9$ |
$2$ |
$(1,5)(2,9)(3,7)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,2,3)(4,6,8)(5,9,7)$ |
$3 \zeta_{3}$ |
$-3 \zeta_{3} - 3$ |
| $1$ |
$3$ |
$(1,3,2)(4,8,6)(5,7,9)$ |
$-3 \zeta_{3} - 3$ |
$3 \zeta_{3}$ |
| $6$ |
$3$ |
$(1,6,9)(2,8,7)(3,4,5)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,4,9)(2,6,7)(3,8,5)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(4,8,6)(5,9,7)$ |
$0$ |
$0$ |
| $6$ |
$3$ |
$(1,9,8)(2,7,4)(3,5,6)$ |
$0$ |
$0$ |
| $9$ |
$6$ |
$(1,7,2,5,3,9)(4,8,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
| $9$ |
$6$ |
$(1,9,3,5,2,7)(4,6,8)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
