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:
| $r_{ 1 }$ |
$=$ |
$ 28 a + 16 + \left(10 a + 23\right)\cdot 31 + \left(14 a + 5\right)\cdot 31^{2} + \left(3 a + 23\right)\cdot 31^{3} + \left(25 a + 10\right)\cdot 31^{4} + \left(26 a + 5\right)\cdot 31^{5} + \left(4 a + 18\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 8\cdot 31 + 6\cdot 31^{2} + 27\cdot 31^{3} + 9\cdot 31^{4} + 13\cdot 31^{5} + 9\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 17 + \left(11 a + 16\right)\cdot 31 + \left(24 a + 14\right)\cdot 31^{2} + \left(21 a + 2\right)\cdot 31^{3} + \left(a + 2\right)\cdot 31^{4} + \left(11 a + 19\right)\cdot 31^{5} + \left(18 a + 30\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + 4 + \left(19 a + 30\right)\cdot 31 + \left(6 a + 20\right)\cdot 31^{2} + \left(9 a + 21\right)\cdot 31^{3} + \left(29 a + 14\right)\cdot 31^{4} + \left(19 a + 8\right)\cdot 31^{5} + \left(12 a + 25\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 28\cdot 31 + 21\cdot 31^{2} + 2\cdot 31^{3} + 29\cdot 31^{4} + 12\cdot 31^{5} + 8\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 3 a + 10 + \left(20 a + 17\right)\cdot 31 + \left(16 a + 23\right)\cdot 31^{2} + \left(27 a + 15\right)\cdot 31^{3} + \left(5 a + 26\right)\cdot 31^{4} + \left(4 a + 2\right)\cdot 31^{5} + \left(26 a + 1\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)$ |
| $(1,2)(5,6)$ |
| $(1,5,4)(2,3,6)$ |
| $(1,2,3)(4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,6)(2,5)$ |
$-1$ |
| $6$ |
$2$ |
$(1,5)(2,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,4,5)(2,6,3)$ |
$0$ |
| $6$ |
$4$ |
$(1,6)(2,3,5,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
