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