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