Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 16.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20 a + 16 + \left(26 a + 1\right)\cdot 41 + \left(21 a + 38\right)\cdot 41^{2} + \left(18 a + 25\right)\cdot 41^{3} + \left(34 a + 20\right)\cdot 41^{4} + \left(26 a + 40\right)\cdot 41^{5} + \left(17 a + 34\right)\cdot 41^{6} + \left(29 a + 19\right)\cdot 41^{7} + \left(21 a + 4\right)\cdot 41^{8} + \left(37 a + 19\right)\cdot 41^{9} + \left(38 a + 35\right)\cdot 41^{10} + \left(17 a + 5\right)\cdot 41^{11} + \left(27 a + 1\right)\cdot 41^{12} + 30\cdot 41^{13} + \left(10 a + 22\right)\cdot 41^{14} + \left(10 a + 34\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 37 a + 32 + \left(20 a + 24\right)\cdot 41 + \left(14 a + 26\right)\cdot 41^{2} + \left(39 a + 31\right)\cdot 41^{3} + \left(16 a + 21\right)\cdot 41^{4} + \left(16 a + 10\right)\cdot 41^{5} + \left(25 a + 7\right)\cdot 41^{6} + \left(27 a + 12\right)\cdot 41^{7} + \left(22 a + 39\right)\cdot 41^{8} + 28 a\cdot 41^{9} + \left(7 a + 14\right)\cdot 41^{10} + \left(13 a + 34\right)\cdot 41^{11} + \left(2 a + 15\right)\cdot 41^{12} + \left(28 a + 7\right)\cdot 41^{13} + \left(40 a + 7\right)\cdot 41^{14} + \left(37 a + 27\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 2 + 4\cdot 41 + 35\cdot 41^{2} + 4\cdot 41^{3} + 19\cdot 41^{4} + 11\cdot 41^{6} + 16\cdot 41^{7} + 31\cdot 41^{8} + 34\cdot 41^{9} + 25\cdot 41^{10} + 4\cdot 41^{11} + 8\cdot 41^{12} + 17\cdot 41^{13} + 22\cdot 41^{14} + 13\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 20 + 22\cdot 41 + 20\cdot 41^{2} + 29\cdot 41^{3} + 4\cdot 41^{4} + 24\cdot 41^{5} + 23\cdot 41^{6} + 37\cdot 41^{7} + 9\cdot 41^{8} + 17\cdot 41^{9} + 6\cdot 41^{10} + 32\cdot 41^{11} + 22\cdot 41^{12} + 15\cdot 41^{13} + 40\cdot 41^{14} + 14\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 21 a + 35 + \left(14 a + 19\right)\cdot 41 + \left(19 a + 35\right)\cdot 41^{2} + \left(22 a + 18\right)\cdot 41^{3} + \left(6 a + 23\right)\cdot 41^{4} + \left(14 a + 4\right)\cdot 41^{5} + \left(23 a + 20\right)\cdot 41^{6} + \left(11 a + 8\right)\cdot 41^{7} + \left(19 a + 40\right)\cdot 41^{8} + \left(3 a + 27\right)\cdot 41^{9} + \left(2 a + 32\right)\cdot 41^{10} + \left(23 a + 20\right)\cdot 41^{11} + \left(13 a + 24\right)\cdot 41^{12} + \left(40 a + 4\right)\cdot 41^{13} + \left(30 a + 11\right)\cdot 41^{14} + \left(30 a + 14\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 4 a + 20 + \left(20 a + 9\right)\cdot 41 + \left(26 a + 8\right)\cdot 41^{2} + \left(a + 12\right)\cdot 41^{3} + \left(24 a + 33\right)\cdot 41^{4} + \left(24 a + 1\right)\cdot 41^{5} + \left(15 a + 26\right)\cdot 41^{6} + \left(13 a + 28\right)\cdot 41^{7} + \left(18 a + 38\right)\cdot 41^{8} + \left(12 a + 22\right)\cdot 41^{9} + \left(33 a + 8\right)\cdot 41^{10} + \left(27 a + 25\right)\cdot 41^{11} + \left(38 a + 9\right)\cdot 41^{12} + \left(12 a + 7\right)\cdot 41^{13} + 19\cdot 41^{14} + \left(3 a + 18\right)\cdot 41^{15} +O\left(41^{ 16 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(1,5,3)(2,6,4)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(5,6)$ | $1$ |
| $3$ | $2$ | $(3,4)(5,6)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $6$ | $2$ | $(1,3)(2,4)(5,6)$ | $1$ |
| $8$ | $3$ | $(1,5,3)(2,6,4)$ | $0$ |
| $6$ | $4$ | $(3,5,4,6)$ | $-1$ |
| $6$ | $4$ | $(1,2)(3,5,4,6)$ | $1$ |
| $8$ | $6$ | $(1,5,4,2,6,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
