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