Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 a + 15 + \left(19 a + 19\right)\cdot 47 + \left(44 a + 30\right)\cdot 47^{2} + \left(25 a + 4\right)\cdot 47^{3} + \left(23 a + 26\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 a + 30 + \left(18 a + 27\right)\cdot 47 + \left(37 a + 15\right)\cdot 47^{2} + \left(6 a + 34\right)\cdot 47^{3} + \left(28 a + 31\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 32 a + 45 + \left(27 a + 42\right)\cdot 47 + \left(2 a + 5\right)\cdot 47^{2} + \left(21 a + 12\right)\cdot 47^{3} + 23 a\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 + 16\cdot 47 + 6\cdot 47^{2} + 2\cdot 47^{3} + 28\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 a + 11 + \left(28 a + 3\right)\cdot 47 + \left(9 a + 25\right)\cdot 47^{2} + \left(40 a + 10\right)\cdot 47^{3} + \left(18 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 31\cdot 47 + 10\cdot 47^{2} + 30\cdot 47^{3} + 20\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)$ |
| $(1,3)$ |
| $(1,3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $6$ | $2$ | $(1,3)$ | $-2$ |
| $9$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $4$ | $3$ | $(1,3,6)(2,4,5)$ | $-2$ |
| $4$ | $3$ | $(2,4,5)$ | $1$ |
| $18$ | $4$ | $(1,4,3,2)(5,6)$ | $0$ |
| $12$ | $6$ | $(1,2,3,4,6,5)$ | $0$ |
| $12$ | $6$ | $(1,3)(2,4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
