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:
| $r_{ 1 }$ |
$=$ |
$ 42 a + 36 + \left(11 a + 8\right)\cdot 47 + \left(32 a + 35\right)\cdot 47^{2} + 3\cdot 47^{3} + \left(16 a + 3\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 a + 26 + \left(35 a + 37\right)\cdot 47 + \left(14 a + 40\right)\cdot 47^{2} + \left(46 a + 19\right)\cdot 47^{3} + \left(30 a + 34\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 12 + \left(35 a + 38\right)\cdot 47 + \left(14 a + 11\right)\cdot 47^{2} + \left(46 a + 43\right)\cdot 47^{3} + \left(30 a + 43\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 44\cdot 47 + 16\cdot 47^{2} + 7\cdot 47^{3} + 25\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 42 a + 22 + \left(11 a + 9\right)\cdot 47 + \left(32 a + 6\right)\cdot 47^{2} + 27\cdot 47^{3} + \left(16 a + 12\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 6 + 2\cdot 47 + 30\cdot 47^{2} + 39\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 |
| $(2,6)(4,5)$ |
| $(1,3)(2,5)(4,6)$ |
| $(1,2)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-2$ |
| $3$ | $2$ | $(1,2)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,5)(2,3)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,6,2)(3,4,5)$ | $-1$ |
| $2$ | $6$ | $(1,4,2,3,6,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
