Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 a + 24 + \left(35 a + 23\right)\cdot 47 + \left(37 a + 14\right)\cdot 47^{2} + \left(15 a + 38\right)\cdot 47^{3} + \left(45 a + 15\right)\cdot 47^{4} + \left(27 a + 2\right)\cdot 47^{5} + 16\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 12 + \left(14 a + 9\right)\cdot 47 + \left(37 a + 10\right)\cdot 47^{2} + \left(9 a + 44\right)\cdot 47^{3} + 40\cdot 47^{4} + \left(43 a + 43\right)\cdot 47^{5} + \left(24 a + 21\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 a + 5 + \left(32 a + 18\right)\cdot 47 + \left(9 a + 23\right)\cdot 47^{2} + \left(37 a + 26\right)\cdot 47^{3} + \left(46 a + 31\right)\cdot 47^{4} + \left(3 a + 35\right)\cdot 47^{5} + \left(22 a + 28\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 a + 1 + \left(11 a + 35\right)\cdot 47 + \left(9 a + 7\right)\cdot 47^{2} + \left(31 a + 32\right)\cdot 47^{3} + \left(a + 43\right)\cdot 47^{4} + \left(19 a + 12\right)\cdot 47^{5} + \left(46 a + 36\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 8 a + 18 + \left(26 a + 5\right)\cdot 47 + \left(46 a + 9\right)\cdot 47^{2} + \left(40 a + 29\right)\cdot 47^{3} + \left(a + 46\right)\cdot 47^{4} + \left(15 a + 8\right)\cdot 47^{5} + \left(24 a + 2\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 39 a + 34 + \left(20 a + 2\right)\cdot 47 + 29\cdot 47^{2} + \left(6 a + 17\right)\cdot 47^{3} + \left(45 a + 9\right)\cdot 47^{4} + \left(31 a + 37\right)\cdot 47^{5} + \left(22 a + 35\right)\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5,3)(2,4,6)$ |
| $(1,6)(2,5)(3,4)$ |
| $(2,6,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $3$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $1$ | $3$ | $(1,5,3)(2,4,6)$ | $2 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,5)(2,6,4)$ | $-2 \zeta_{3} - 2$ |
| $2$ | $3$ | $(2,6,4)$ | $-\zeta_{3}$ |
| $2$ | $3$ | $(2,4,6)$ | $\zeta_{3} + 1$ |
| $2$ | $3$ | $(1,3,5)(2,4,6)$ | $-1$ |
| $3$ | $6$ | $(1,2,3,6,5,4)$ | $0$ |
| $3$ | $6$ | $(1,4,5,6,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
