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^{3} + x + 28 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 a^{2} + \left(15 a^{2} + 29 a + 18\right)\cdot 31 + \left(7 a^{2} + 26 a + 9\right)\cdot 31^{2} + \left(17 a^{2} + 18\right)\cdot 31^{3} + \left(13 a^{2} + 26 a + 11\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a^{2} + 16 a + 26 + \left(19 a^{2} + 28 a + 20\right)\cdot 31 + \left(4 a^{2} + 15 a + 7\right)\cdot 31^{2} + \left(14 a^{2} + 26 a + 16\right)\cdot 31^{3} + \left(15 a^{2} + 23 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a^{2} + 7 a + 19 + \left(a^{2} + 26 a + 24\right)\cdot 31 + \left(27 a^{2} + 14 a + 23\right)\cdot 31^{2} + \left(30 a^{2} + 24 a + 13\right)\cdot 31^{3} + \left(22 a^{2} + 23 a + 2\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a^{2} + 15 a + 4 + \left(26 a^{2} + 4 a + 15\right)\cdot 31 + \left(18 a^{2} + 19 a + 27\right)\cdot 31^{2} + \left(30 a^{2} + 3 a + 16\right)\cdot 31^{3} + \left(a^{2} + 12 a + 24\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 20 a^{2} + 30 a + 4 + \left(3 a^{2} + 27 a + 5\right)\cdot 31 + \left(13 a^{2} + 10 a + 4\right)\cdot 31^{2} + \left(2 a^{2} + 17 a + 5\right)\cdot 31^{3} + \left(30 a^{2} + 6 a + 7\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 a^{2} + 25 a + 11 + \left(25 a^{2} + 7 a + 9\right)\cdot 31 + \left(21 a^{2} + 5 a + 20\right)\cdot 31^{2} + \left(28 a^{2} + 20 a + 22\right)\cdot 31^{3} + \left(8 a^{2} + 13\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)$ |
| $(1,5,6,3,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$5$ |
| $10$ |
$2$ |
$(1,5)(2,6)(3,4)$ |
$-1$ |
| $15$ |
$2$ |
$(1,5)(3,6)$ |
$1$ |
| $20$ |
$3$ |
$(1,6,2)(3,4,5)$ |
$-1$ |
| $30$ |
$4$ |
$(1,3,5,6)$ |
$1$ |
| $24$ |
$5$ |
$(2,5,4,6,3)$ |
$0$ |
| $20$ |
$6$ |
$(1,5,6,3,2,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
