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:
| $r_{ 1 }$ |
$=$ |
$ 32 a + 15 + \left(22 a + 40\right)\cdot 47 + \left(16 a + 41\right)\cdot 47^{2} + \left(15 a + 39\right)\cdot 47^{3} + \left(39 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 5\cdot 47 + 28\cdot 47^{2} + 10\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 a + 30 + 31 a\cdot 47 + \left(31 a + 31\right)\cdot 47^{2} + \left(4 a + 34\right)\cdot 47^{3} + \left(18 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 a + 32 + \left(24 a + 6\right)\cdot 47 + \left(30 a + 5\right)\cdot 47^{2} + \left(31 a + 7\right)\cdot 47^{3} + \left(7 a + 8\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 41\cdot 47 + 18\cdot 47^{2} + 36\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 30 a + 17 + \left(15 a + 46\right)\cdot 47 + \left(15 a + 15\right)\cdot 47^{2} + \left(42 a + 12\right)\cdot 47^{3} + \left(28 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,5)(3,6)$ |
| $(1,5,6)(2,3,4)$ |
| $(1,4)(3,6)$ |
| $(1,5,4,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(2,5)(3,6)$ | $-1$ |
| $6$ | $2$ | $(1,2)(3,6)(4,5)$ | $-1$ |
| $8$ | $3$ | $(1,5,6)(2,3,4)$ | $0$ |
| $6$ | $4$ | $(1,5,4,2)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
