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:
| $r_{ 1 }$ |
$=$ |
$ 5 + 6\cdot 47 + 31\cdot 47^{2} + 3\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 11 + \left(a + 21\right)\cdot 47 + \left(38 a + 24\right)\cdot 47^{2} + \left(32 a + 44\right)\cdot 47^{3} + \left(45 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 24 + \left(45 a + 41\right)\cdot 47 + \left(8 a + 4\right)\cdot 47^{2} + \left(14 a + 25\right)\cdot 47^{3} + \left(a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 38 a + 1 + 41\cdot 47 + \left(39 a + 35\right)\cdot 47^{2} + \left(10 a + 23\right)\cdot 47^{3} + \left(43 a + 44\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 26\cdot 47 + 25\cdot 47^{2} + 37\cdot 47^{3} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 9 a + 30 + \left(46 a + 4\right)\cdot 47 + \left(7 a + 19\right)\cdot 47^{2} + \left(36 a + 6\right)\cdot 47^{3} + \left(3 a + 26\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4)(2,5)(3,6)$ |
| $(1,2)$ |
| $(1,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $6$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $6$ | $2$ | $(2,3)$ | $2$ |
| $9$ | $2$ | $(2,3)(5,6)$ | $0$ |
| $4$ | $3$ | $(1,2,3)$ | $1$ |
| $4$ | $3$ | $(1,2,3)(4,5,6)$ | $-2$ |
| $18$ | $4$ | $(1,4)(2,6,3,5)$ | $0$ |
| $12$ | $6$ | $(1,5,2,6,3,4)$ | $0$ |
| $12$ | $6$ | $(2,3)(4,5,6)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
