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