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