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