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 a + 14 + \left(13 a + 46\right)\cdot 47 + \left(9 a + 41\right)\cdot 47^{2} + \left(31 a + 30\right)\cdot 47^{3} + \left(5 a + 33\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 33 a + 42 + \left(33 a + 11\right)\cdot 47 + 37 a\cdot 47^{2} + \left(15 a + 37\right)\cdot 47^{3} + \left(41 a + 13\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 a + 11 + \left(22 a + 27\right)\cdot 47 + \left(45 a + 17\right)\cdot 47^{2} + \left(17 a + 7\right)\cdot 47^{3} + \left(24 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 a + 42 + \left(24 a + 33\right)\cdot 47 + \left(a + 38\right)\cdot 47^{2} + \left(29 a + 44\right)\cdot 47^{3} + \left(22 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 + 21\cdot 47 + 42\cdot 47^{2} + 20\cdot 47^{3} + 33\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 |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
