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