Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 10 + 10 a\cdot 47 + \left(33 a + 9\right)\cdot 47^{2} + \left(38 a + 30\right)\cdot 47^{3} + \left(38 a + 7\right)\cdot 47^{4} + 46\cdot 47^{5} + \left(a + 35\right)\cdot 47^{6} + \left(35 a + 29\right)\cdot 47^{7} + \left(42 a + 24\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 36\cdot 47 + 26\cdot 47^{2} + 47^{3} + 12\cdot 47^{4} + 2\cdot 47^{5} + 4\cdot 47^{6} + 21\cdot 47^{7} +O\left(47^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 28 + 10\cdot 47 + 11\cdot 47^{2} + 10\cdot 47^{3} + 27\cdot 47^{4} + 47^{5} + 30\cdot 47^{6} + 16\cdot 47^{7} + 29\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 41 a + 22 + \left(36 a + 14\right)\cdot 47 + \left(13 a + 18\right)\cdot 47^{2} + \left(8 a + 27\right)\cdot 47^{3} + \left(8 a + 46\right)\cdot 47^{4} + \left(46 a + 8\right)\cdot 47^{5} + \left(45 a + 37\right)\cdot 47^{6} + \left(11 a + 4\right)\cdot 47^{7} + \left(4 a + 28\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 9 + \left(6 a + 37\right)\cdot 47 + \left(39 a + 1\right)\cdot 47^{2} + \left(13 a + 18\right)\cdot 47^{3} + \left(8 a + 22\right)\cdot 47^{4} + \left(20 a + 1\right)\cdot 47^{5} + \left(12 a + 38\right)\cdot 47^{6} + \left(43 a + 20\right)\cdot 47^{7} + \left(39 a + 34\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 23 + \left(40 a + 42\right)\cdot 47 + \left(7 a + 26\right)\cdot 47^{2} + \left(33 a + 6\right)\cdot 47^{3} + \left(38 a + 25\right)\cdot 47^{4} + \left(26 a + 33\right)\cdot 47^{5} + \left(34 a + 42\right)\cdot 47^{6} + 3 a\cdot 47^{7} + \left(7 a + 24\right)\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(5,6)$ |
| $(1,6)$ |
| $(1,4,2)(3,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(2,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,6)(2,3)$ |
$-1$ |
| $6$ |
$2$ |
$(1,4)(5,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,4)(2,3)(5,6)$ |
$-1$ |
| $8$ |
$3$ |
$(1,4,2)(3,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,6,3)$ |
$1$ |
| $6$ |
$4$ |
$(1,6)(2,5,3,4)$ |
$-1$ |
| $8$ |
$6$ |
$(1,4,2,6,5,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
