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