Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a + 12 + 25 a\cdot 31 + \left(15 a + 3\right)\cdot 31^{2} + \left(3 a + 5\right)\cdot 31^{3} + \left(29 a + 18\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 15 a + 13 + \left(5 a + 4\right)\cdot 31 + \left(15 a + 9\right)\cdot 31^{2} + \left(27 a + 27\right)\cdot 31^{3} + \left(a + 10\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 23 + \left(19 a + 29\right)\cdot 31 + \left(23 a + 5\right)\cdot 31^{2} + \left(28 a + 1\right)\cdot 31^{3} + \left(4 a + 26\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 21 + \left(11 a + 8\right)\cdot 31 + \left(7 a + 2\right)\cdot 31^{2} + \left(2 a + 4\right)\cdot 31^{3} + \left(26 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 6 + 26\cdot 31 + 18\cdot 31^{2} + 29\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 18 + 23\cdot 31 + 22\cdot 31^{2} + 25\cdot 31^{3} + 28\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)$ |
| $(1,2)$ |
| $(1,2,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $2$ |
| $6$ | $2$ | $(2,5)$ | $0$ |
| $9$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $1$ |
| $4$ | $3$ | $(3,4,6)$ | $-2$ |
| $18$ | $4$ | $(1,3)(2,6,5,4)$ | $0$ |
| $12$ | $6$ | $(1,3,2,4,5,6)$ | $-1$ |
| $12$ | $6$ | $(2,5)(3,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
