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:
| $r_{ 1 }$ |
$=$ |
$ 27 + 20\cdot 31 + 29\cdot 31^{2} + 10\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 a + 9 + \left(28 a + 19\right)\cdot 31 + \left(12 a + 17\right)\cdot 31^{2} + \left(6 a + 25\right)\cdot 31^{3} + \left(16 a + 9\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 31 + 8\cdot 31^{2} + 10\cdot 31^{3} + 19\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 17 + \left(8 a + 9\right)\cdot 31 + 15 a\cdot 31^{2} + \left(28 a + 5\right)\cdot 31^{3} + \left(5 a + 14\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 26 + \left(2 a + 21\right)\cdot 31 + \left(18 a + 14\right)\cdot 31^{2} + \left(24 a + 25\right)\cdot 31^{3} + \left(14 a + 4\right)\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 29 + \left(22 a + 19\right)\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(2 a + 15\right)\cdot 31^{3} + \left(25 a + 28\right)\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$0$ |
| $6$ |
$2$ |
$(2,5)$ |
$-2$ |
| $9$ |
$2$ |
$(2,5)(4,6)$ |
$0$ |
| $4$ |
$3$ |
$(1,2,5)$ |
$1$ |
| $4$ |
$3$ |
$(1,2,5)(3,4,6)$ |
$-2$ |
| $18$ |
$4$ |
$(1,3)(2,6,5,4)$ |
$0$ |
| $12$ |
$6$ |
$(1,4,2,6,5,3)$ |
$0$ |
| $12$ |
$6$ |
$(2,5)(3,4,6)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
