Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 a + \left(13 a + 26\right)\cdot 31 + \left(12 a + 30\right)\cdot 31^{2} + \left(27 a + 14\right)\cdot 31^{3} + \left(27 a + 23\right)\cdot 31^{4} + \left(12 a + 30\right)\cdot 31^{5} + \left(29 a + 23\right)\cdot 31^{6} + \left(2 a + 23\right)\cdot 31^{7} + \left(8 a + 21\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 a + 19 + \left(17 a + 28\right)\cdot 31 + \left(18 a + 10\right)\cdot 31^{2} + \left(3 a + 26\right)\cdot 31^{3} + \left(3 a + 20\right)\cdot 31^{4} + \left(18 a + 28\right)\cdot 31^{5} + \left(a + 7\right)\cdot 31^{6} + 28 a\cdot 31^{7} + \left(22 a + 4\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 7\cdot 31 + 20\cdot 31^{2} + 20\cdot 31^{3} + 17\cdot 31^{4} + 2\cdot 31^{5} + 30\cdot 31^{6} + 6\cdot 31^{7} + 5\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 a + 29 + \left(25 a + 5\right)\cdot 31 + \left(30 a + 24\right)\cdot 31^{2} + 19\cdot 31^{3} + \left(13 a + 6\right)\cdot 31^{4} + \left(13 a + 24\right)\cdot 31^{5} + \left(27 a + 16\right)\cdot 31^{6} + \left(29 a + 24\right)\cdot 31^{7} + \left(14 a + 14\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 4 + 15\cdot 31 + 8\cdot 31^{2} + 20\cdot 31^{3} + 23\cdot 31^{4} + 30\cdot 31^{5} + 17\cdot 31^{6} + 11\cdot 31^{7} + 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a + 30 + \left(5 a + 9\right)\cdot 31 + 29\cdot 31^{2} + \left(30 a + 21\right)\cdot 31^{3} + 17 a\cdot 31^{4} + \left(17 a + 7\right)\cdot 31^{5} + \left(3 a + 27\right)\cdot 31^{6} + \left(a + 25\right)\cdot 31^{7} + \left(16 a + 14\right)\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,3)(4,5)$ |
| $(1,2)(4,6)$ |
| $(1,4)(2,6)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,5)$ |
$-2$ |
| $3$ |
$2$ |
$(1,2)(4,6)$ |
$0$ |
| $3$ |
$2$ |
$(1,4)(2,6)(3,5)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,6,5)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,2,6,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
