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