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