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