Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 45\cdot 47 + 15\cdot 47^{2} + 5\cdot 47^{3} + 43\cdot 47^{4} + 13\cdot 47^{5} + 36\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 a + 13 + \left(3 a + 37\right)\cdot 47 + \left(23 a + 13\right)\cdot 47^{2} + \left(11 a + 11\right)\cdot 47^{3} + \left(11 a + 24\right)\cdot 47^{4} + \left(25 a + 40\right)\cdot 47^{5} + \left(25 a + 2\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 a + 27 + \left(24 a + 6\right)\cdot 47 + \left(7 a + 19\right)\cdot 47^{2} + \left(28 a + 6\right)\cdot 47^{3} + \left(34 a + 6\right)\cdot 47^{4} + \left(18 a + 6\right)\cdot 47^{5} + \left(2 a + 9\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 a + 2 + \left(22 a + 44\right)\cdot 47 + \left(39 a + 9\right)\cdot 47^{2} + \left(18 a + 8\right)\cdot 47^{3} + 12 a\cdot 47^{4} + \left(28 a + 9\right)\cdot 47^{5} + \left(44 a + 42\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 29 a + 2 + \left(43 a + 26\right)\cdot 47 + \left(23 a + 9\right)\cdot 47^{2} + \left(35 a + 11\right)\cdot 47^{3} + \left(35 a + 35\right)\cdot 47^{4} + \left(21 a + 32\right)\cdot 47^{5} + \left(21 a + 28\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 28\cdot 47 + 25\cdot 47^{2} + 4\cdot 47^{3} + 32\cdot 47^{4} + 38\cdot 47^{5} + 21\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-3$ |
| $3$ | $2$ | $(1,6)(3,4)$ | $-1$ |
| $3$ | $2$ | $(1,6)$ | $1$ |
| $6$ | $2$ | $(1,3)(4,6)$ | $1$ |
| $6$ | $2$ | $(1,6)(2,3)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)(4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,6,3)$ | $1$ |
| $6$ | $4$ | $(1,4,6,3)(2,5)$ | $-1$ |
| $8$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
