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