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:
| $r_{ 1 }$ |
$=$ |
$ 31 + 45\cdot 47 + 30\cdot 47^{2} + 2\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 + 5\cdot 47 + 46\cdot 47^{2} + 36\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 a + 8 + \left(20 a + 28\right)\cdot 47 + \left(39 a + 41\right)\cdot 47^{2} + \left(44 a + 44\right)\cdot 47^{3} + \left(25 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 36 + 11\cdot 47 + 16\cdot 47^{2} + 8\cdot 47^{3} + 3\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 46 + \left(26 a + 2\right)\cdot 47 + \left(7 a + 6\right)\cdot 47^{2} + \left(2 a + 1\right)\cdot 47^{3} + \left(21 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $10$ | $2$ | $(1,2)$ | $0$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
