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