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