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