Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $ x^{2} + 45 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4\cdot 47 + 10\cdot 47^{2} + 16\cdot 47^{3} + 11\cdot 47^{4} + 28\cdot 47^{5} + 13\cdot 47^{6} + 24\cdot 47^{7} + 44\cdot 47^{8} + 17\cdot 47^{9} + 44\cdot 47^{10} + 7\cdot 47^{11} + 6\cdot 47^{12} + 35\cdot 47^{13} + 31\cdot 47^{14} +O\left(47^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 1 + 43\cdot 47 + 36\cdot 47^{2} + 30\cdot 47^{3} + 35\cdot 47^{4} + 18\cdot 47^{5} + 33\cdot 47^{6} + 22\cdot 47^{7} + 2\cdot 47^{8} + 29\cdot 47^{9} + 2\cdot 47^{10} + 39\cdot 47^{11} + 40\cdot 47^{12} + 11\cdot 47^{13} + 15\cdot 47^{14} +O\left(47^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 40 a + 40 + \left(32 a + 1\right)\cdot 47 + \left(27 a + 30\right)\cdot 47^{2} + \left(40 a + 10\right)\cdot 47^{3} + \left(8 a + 20\right)\cdot 47^{4} + \left(43 a + 36\right)\cdot 47^{5} + \left(22 a + 6\right)\cdot 47^{6} + \left(30 a + 25\right)\cdot 47^{7} + \left(38 a + 9\right)\cdot 47^{8} + \left(35 a + 38\right)\cdot 47^{9} + \left(11 a + 29\right)\cdot 47^{10} + \left(5 a + 10\right)\cdot 47^{11} + \left(14 a + 21\right)\cdot 47^{12} + \left(3 a + 21\right)\cdot 47^{13} + \left(4 a + 22\right)\cdot 47^{14} +O\left(47^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 7 a + 8 + \left(14 a + 45\right)\cdot 47 + \left(19 a + 16\right)\cdot 47^{2} + \left(6 a + 36\right)\cdot 47^{3} + \left(38 a + 26\right)\cdot 47^{4} + \left(3 a + 10\right)\cdot 47^{5} + \left(24 a + 40\right)\cdot 47^{6} + \left(16 a + 21\right)\cdot 47^{7} + \left(8 a + 37\right)\cdot 47^{8} + \left(11 a + 8\right)\cdot 47^{9} + \left(35 a + 17\right)\cdot 47^{10} + \left(41 a + 36\right)\cdot 47^{11} + \left(32 a + 25\right)\cdot 47^{12} + \left(43 a + 25\right)\cdot 47^{13} + \left(42 a + 24\right)\cdot 47^{14} +O\left(47^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 a + 26 + \left(14 a + 27\right)\cdot 47 + \left(19 a + 5\right)\cdot 47^{2} + \left(6 a + 17\right)\cdot 47^{3} + \left(38 a + 44\right)\cdot 47^{4} + \left(3 a + 19\right)\cdot 47^{5} + \left(24 a + 9\right)\cdot 47^{6} + \left(16 a + 16\right)\cdot 47^{7} + \left(8 a + 9\right)\cdot 47^{8} + \left(11 a + 24\right)\cdot 47^{9} + \left(35 a + 17\right)\cdot 47^{10} + \left(41 a + 9\right)\cdot 47^{11} + \left(32 a + 44\right)\cdot 47^{12} + \left(43 a + 13\right)\cdot 47^{13} + \left(42 a + 27\right)\cdot 47^{14} +O\left(47^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 40 a + 22 + \left(32 a + 19\right)\cdot 47 + \left(27 a + 41\right)\cdot 47^{2} + \left(40 a + 29\right)\cdot 47^{3} + \left(8 a + 2\right)\cdot 47^{4} + \left(43 a + 27\right)\cdot 47^{5} + \left(22 a + 37\right)\cdot 47^{6} + \left(30 a + 30\right)\cdot 47^{7} + \left(38 a + 37\right)\cdot 47^{8} + \left(35 a + 22\right)\cdot 47^{9} + \left(11 a + 29\right)\cdot 47^{10} + \left(5 a + 37\right)\cdot 47^{11} + \left(14 a + 2\right)\cdot 47^{12} + \left(3 a + 33\right)\cdot 47^{13} + \left(4 a + 19\right)\cdot 47^{14} +O\left(47^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(1,2)$ |
| $(1,3,5)(2,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(5,6)$ | $1$ |
| $3$ | $2$ | $(1,2)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,3,5)(2,4,6)$ | $0$ |
| $6$ | $4$ | $(1,5,2,6)$ | $-1$ |
| $6$ | $4$ | $(1,5,2,6)(3,4)$ | $1$ |
| $8$ | $6$ | $(1,3,5,2,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
