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:
| $r_{ 1 }$ |
$=$ |
$ 9 a + 8 + \left(19 a + 13\right)\cdot 31 + \left(12 a + 27\right)\cdot 31^{2} + \left(26 a + 5\right)\cdot 31^{3} + \left(14 a + 1\right)\cdot 31^{4} + 30 a\cdot 31^{5} + \left(a + 8\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 a + 26 + \left(11 a + 11\right)\cdot 31 + \left(18 a + 2\right)\cdot 31^{2} + \left(4 a + 15\right)\cdot 31^{3} + \left(16 a + 4\right)\cdot 31^{4} + 15\cdot 31^{5} + \left(29 a + 12\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 30\cdot 31 + 26\cdot 31^{2} + 9\cdot 31^{3} + 18\cdot 31^{4} + 26\cdot 31^{5} + 4\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 7 + \left(11 a + 26\right)\cdot 31 + \left(18 a + 5\right)\cdot 31^{2} + \left(4 a + 15\right)\cdot 31^{3} + \left(16 a + 19\right)\cdot 31^{4} + 28\cdot 31^{5} + \left(29 a + 5\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 20 + \left(19 a + 27\right)\cdot 31 + \left(12 a + 30\right)\cdot 31^{2} + \left(26 a + 5\right)\cdot 31^{3} + \left(14 a + 16\right)\cdot 31^{4} + \left(30 a + 13\right)\cdot 31^{5} + \left(a + 1\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 7 + 14\cdot 31 + 30\cdot 31^{2} + 9\cdot 31^{3} + 2\cdot 31^{4} + 9\cdot 31^{5} + 29\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(4,5)$ |
| $(2,3)(4,6)$ |
| $(1,4)(2,5)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(2,5)(3,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$-1$ |
| $2$ |
$6$ |
$(1,6,2,5,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
