Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 6.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 21 + 42\cdot 47 + 46\cdot 47^{2} + 10\cdot 47^{3} + 30\cdot 47^{4} + 8\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 24 + \left(36 a + 45\right)\cdot 47 + \left(22 a + 19\right)\cdot 47^{2} + \left(19 a + 21\right)\cdot 47^{3} + \left(46 a + 16\right)\cdot 47^{4} + \left(44 a + 28\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 44 a + 2 + \left(3 a + 34\right)\cdot 47 + \left(27 a + 35\right)\cdot 47^{2} + \left(22 a + 37\right)\cdot 47^{3} + \left(39 a + 38\right)\cdot 47^{4} + \left(36 a + 34\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 a + 43 + \left(43 a + 44\right)\cdot 47 + \left(19 a + 38\right)\cdot 47^{2} + \left(24 a + 8\right)\cdot 47^{3} + \left(7 a + 1\right)\cdot 47^{4} + \left(10 a + 22\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 34 + \left(10 a + 18\right)\cdot 47 + \left(24 a + 29\right)\cdot 47^{2} + \left(27 a + 37\right)\cdot 47^{3} + 42\cdot 47^{4} + \left(2 a + 24\right)\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 2\cdot 47 + 17\cdot 47^{2} + 24\cdot 47^{3} + 11\cdot 47^{4} + 22\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,4)$ |
| $(1,2)(3,6)$ |
| $(1,3)(2,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,6)$ | $0$ |
| $3$ | $2$ | $(1,3)(2,6)(4,5)$ | $0$ |
| $2$ | $3$ | $(1,5,2)(3,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,6,5,3)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
