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:
| $r_{ 1 }$ |
$=$ |
$ 21 a + 16 + \left(35 a + 6\right)\cdot 47 + 41\cdot 47^{2} + \left(a + 24\right)\cdot 47^{3} + \left(8 a + 19\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 21 a + 18 + \left(25 a + 25\right)\cdot 47 + \left(33 a + 23\right)\cdot 47^{2} + \left(18 a + 7\right)\cdot 47^{3} + \left(37 a + 26\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 11 + \left(11 a + 9\right)\cdot 47 + \left(46 a + 7\right)\cdot 47^{2} + \left(45 a + 26\right)\cdot 47^{3} + \left(38 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 a + 13 + \left(21 a + 8\right)\cdot 47 + \left(13 a + 18\right)\cdot 47^{2} + \left(28 a + 11\right)\cdot 47^{3} + \left(9 a + 35\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 37 + 44\cdot 47 + 3\cdot 47^{2} + 24\cdot 47^{3} + 25\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 values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $10$ |
$2$ |
$(1,2)$ |
$2$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$1$ |
| $30$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
| $24$ |
$5$ |
$(1,2,3,4,5)$ |
$-1$ |
| $20$ |
$6$ |
$(1,2,3)(4,5)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
