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:
| $r_{ 1 }$ |
$=$ |
$ 14 + 6\cdot 47 + 23\cdot 47^{2} + 12\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 42 a + 26 + \left(34 a + 26\right)\cdot 47 + \left(23 a + 14\right)\cdot 47^{2} + \left(16 a + 23\right)\cdot 47^{3} + \left(19 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 16 + \left(12 a + 7\right)\cdot 47 + \left(23 a + 27\right)\cdot 47^{2} + \left(30 a + 32\right)\cdot 47^{3} + \left(27 a + 29\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 a + 1 + \left(10 a + 45\right)\cdot 47 + \left(38 a + 8\right)\cdot 47^{2} + \left(6 a + 7\right)\cdot 47^{3} + \left(8 a + 41\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 a + 19 + \left(36 a + 9\right)\cdot 47 + \left(8 a + 28\right)\cdot 47^{2} + \left(40 a + 29\right)\cdot 47^{3} + \left(38 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 20 + 46\cdot 47 + 38\cdot 47^{2} + 35\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3)$ |
| $(1,2)(3,4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $2$ |
| $40$ | $3$ | $(1,2,3)$ | $-1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $-1$ |
| $72$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $72$ | $5$ | $(1,3,4,5,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
