Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 15.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 36 a + 3 + \left(17 a + 18\right)\cdot 41 + \left(34 a + 20\right)\cdot 41^{2} + \left(24 a + 20\right)\cdot 41^{3} + 22\cdot 41^{4} + \left(13 a + 13\right)\cdot 41^{5} + \left(21 a + 4\right)\cdot 41^{6} + \left(36 a + 10\right)\cdot 41^{7} + \left(19 a + 30\right)\cdot 41^{8} + \left(21 a + 7\right)\cdot 41^{9} + \left(30 a + 35\right)\cdot 41^{10} + \left(22 a + 30\right)\cdot 41^{11} + \left(31 a + 7\right)\cdot 41^{12} + \left(27 a + 11\right)\cdot 41^{13} + \left(12 a + 17\right)\cdot 41^{14} +O\left(41^{ 15 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 19 + \left(33 a + 9\right)\cdot 41 + \left(15 a + 27\right)\cdot 41^{2} + \left(24 a + 31\right)\cdot 41^{3} + \left(8 a + 15\right)\cdot 41^{4} + \left(32 a + 1\right)\cdot 41^{5} + \left(2 a + 16\right)\cdot 41^{6} + \left(36 a + 27\right)\cdot 41^{7} + \left(39 a + 18\right)\cdot 41^{8} + \left(9 a + 10\right)\cdot 41^{9} + \left(39 a + 3\right)\cdot 41^{10} + \left(36 a + 22\right)\cdot 41^{11} + \left(17 a + 36\right)\cdot 41^{12} + \left(32 a + 38\right)\cdot 41^{13} + \left(15 a + 14\right)\cdot 41^{14} +O\left(41^{ 15 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 a + 29 + \left(23 a + 35\right)\cdot 41 + \left(6 a + 23\right)\cdot 41^{2} + \left(16 a + 19\right)\cdot 41^{3} + \left(40 a + 40\right)\cdot 41^{4} + \left(27 a + 10\right)\cdot 41^{5} + \left(19 a + 14\right)\cdot 41^{6} + \left(4 a + 16\right)\cdot 41^{7} + \left(21 a + 12\right)\cdot 41^{8} + \left(19 a + 11\right)\cdot 41^{9} + \left(10 a + 23\right)\cdot 41^{10} + \left(18 a + 27\right)\cdot 41^{11} + \left(9 a + 38\right)\cdot 41^{12} + \left(13 a + 21\right)\cdot 41^{13} + \left(28 a + 27\right)\cdot 41^{14} +O\left(41^{ 15 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 14\cdot 41 + 30\cdot 41^{2} + 7\cdot 41^{3} + 14\cdot 41^{4} + 11\cdot 41^{5} + 23\cdot 41^{6} + 7\cdot 41^{7} + 38\cdot 41^{8} + 13\cdot 41^{9} + 26\cdot 41^{10} + 33\cdot 41^{11} + 33\cdot 41^{12} + 35\cdot 41^{13} + 21\cdot 41^{14} +O\left(41^{ 15 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 40 a + 22 + \left(7 a + 25\right)\cdot 41 + 25 a\cdot 41^{2} + \left(16 a + 7\right)\cdot 41^{3} + \left(32 a + 17\right)\cdot 41^{4} + \left(8 a + 7\right)\cdot 41^{5} + \left(38 a + 33\right)\cdot 41^{6} + \left(4 a + 9\right)\cdot 41^{7} + \left(a + 20\right)\cdot 41^{8} + 31 a\cdot 41^{9} + \left(a + 29\right)\cdot 41^{10} + \left(4 a + 11\right)\cdot 41^{11} + \left(23 a + 12\right)\cdot 41^{12} + \left(8 a + 36\right)\cdot 41^{13} + \left(25 a + 29\right)\cdot 41^{14} +O\left(41^{ 15 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 16 + 19\cdot 41 + 20\cdot 41^{2} + 36\cdot 41^{3} + 12\cdot 41^{4} + 37\cdot 41^{5} + 31\cdot 41^{6} + 10\cdot 41^{7} + 3\cdot 41^{8} + 38\cdot 41^{9} + 5\cdot 41^{10} + 38\cdot 41^{11} + 34\cdot 41^{12} + 19\cdot 41^{13} + 11\cdot 41^{14} +O\left(41^{ 15 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3,4)(2,5,6)$ |
| $(4,6)$ |
| $(1,4)(2,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-3$ |
| $3$ | $2$ | $(3,5)$ | $1$ |
| $3$ | $2$ | $(3,5)(4,6)$ | $-1$ |
| $6$ | $2$ | $(1,4)(2,6)$ | $1$ |
| $6$ | $2$ | $(1,4)(2,6)(3,5)$ | $-1$ |
| $8$ | $3$ | $(1,3,4)(2,5,6)$ | $0$ |
| $6$ | $4$ | $(3,6,5,4)$ | $1$ |
| $6$ | $4$ | $(1,2)(3,6,5,4)$ | $-1$ |
| $8$ | $6$ | $(1,3,6,2,5,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
