Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{2} + 29 x + 3 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2\cdot 31 + 7\cdot 31^{2} + 10\cdot 31^{3} + 13\cdot 31^{4} + 12\cdot 31^{5} + 5\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 14\cdot 31 + 11\cdot 31^{3} + 11\cdot 31^{4} + 30\cdot 31^{5} + 25\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 a + 19 + \left(27 a + 28\right)\cdot 31 + 21 a\cdot 31^{2} + \left(6 a + 14\right)\cdot 31^{3} + \left(28 a + 16\right)\cdot 31^{4} + \left(25 a + 30\right)\cdot 31^{5} + \left(9 a + 15\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 a + \left(3 a + 25\right)\cdot 31 + \left(9 a + 10\right)\cdot 31^{2} + \left(24 a + 6\right)\cdot 31^{3} + \left(2 a + 2\right)\cdot 31^{4} + \left(5 a + 10\right)\cdot 31^{5} + \left(21 a + 30\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 9 a + 13 + \left(27 a + 10\right)\cdot 31 + \left(21 a + 25\right)\cdot 31^{2} + \left(6 a + 14\right)\cdot 31^{3} + \left(28 a + 14\right)\cdot 31^{4} + \left(25 a + 17\right)\cdot 31^{5} + \left(9 a + 5\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 a + 6 + \left(3 a + 12\right)\cdot 31 + \left(9 a + 17\right)\cdot 31^{2} + \left(24 a + 5\right)\cdot 31^{3} + \left(2 a + 4\right)\cdot 31^{4} + \left(5 a + 23\right)\cdot 31^{5} + \left(21 a + 9\right)\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)$ |
| $(3,6)(4,5)$ |
| $(1,3,6)(2,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)$ |
$-2$ |
| $3$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $3$ |
$2$ |
$(1,2)(3,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,6)(2,5,4)$ |
$-1$ |
| $2$ |
$6$ |
$(1,5,6,2,3,4)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
