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