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