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