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:
| $r_{ 1 }$ |
$=$ |
$ 40 a + 12 + 41 + \left(25 a + 23\right)\cdot 41^{2} + \left(14 a + 6\right)\cdot 41^{3} + \left(11 a + 19\right)\cdot 41^{4} + \left(19 a + 1\right)\cdot 41^{5} + \left(a + 7\right)\cdot 41^{6} + \left(29 a + 19\right)\cdot 41^{7} + \left(a + 4\right)\cdot 41^{8} + \left(11 a + 9\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ a + 9 + \left(40 a + 5\right)\cdot 41 + \left(15 a + 15\right)\cdot 41^{2} + \left(26 a + 25\right)\cdot 41^{3} + \left(29 a + 38\right)\cdot 41^{4} + \left(21 a + 6\right)\cdot 41^{5} + \left(39 a + 33\right)\cdot 41^{6} + \left(11 a + 22\right)\cdot 41^{7} + \left(39 a + 21\right)\cdot 41^{8} + \left(29 a + 40\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 a + 8 + \left(39 a + 15\right)\cdot 41 + \left(37 a + 36\right)\cdot 41^{2} + \left(11 a + 20\right)\cdot 41^{3} + \left(39 a + 6\right)\cdot 41^{4} + \left(2 a + 20\right)\cdot 41^{5} + \left(2 a + 14\right)\cdot 41^{6} + \left(39 a + 11\right)\cdot 41^{7} + \left(13 a + 6\right)\cdot 41^{8} + \left(35 a + 27\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 26\cdot 41 + 24\cdot 41^{2} + 11\cdot 41^{3} + 24\cdot 41^{4} + 9\cdot 41^{5} + 19\cdot 41^{6} + 27\cdot 41^{7} + 38\cdot 41^{8} + 6\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 a + 6 + \left(a + 38\right)\cdot 41 + \left(3 a + 28\right)\cdot 41^{2} + \left(29 a + 18\right)\cdot 41^{3} + \left(a + 30\right)\cdot 41^{4} + \left(38 a + 30\right)\cdot 41^{5} + \left(38 a + 17\right)\cdot 41^{6} + \left(a + 3\right)\cdot 41^{7} + \left(27 a + 9\right)\cdot 41^{8} + \left(5 a + 37\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 12\cdot 41 + 38\cdot 41^{2} + 19\cdot 41^{3} + 19\cdot 41^{4} + 12\cdot 41^{5} + 11\cdot 41^{6} + 2\cdot 41^{7} + 20\cdot 41^{8} + 12\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 7 a + \left(6 a + 6\right)\cdot 41 + \left(27 a + 2\right)\cdot 41^{2} + \left(13 a + 3\right)\cdot 41^{3} + \left(27 a + 40\right)\cdot 41^{4} + \left(17 a + 7\right)\cdot 41^{5} + \left(7 a + 28\right)\cdot 41^{6} + \left(7 a + 31\right)\cdot 41^{7} + \left(15 a + 12\right)\cdot 41^{8} + \left(14 a + 1\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 34 a + 21 + \left(34 a + 17\right)\cdot 41 + \left(13 a + 36\right)\cdot 41^{2} + \left(27 a + 16\right)\cdot 41^{3} + \left(13 a + 26\right)\cdot 41^{4} + \left(23 a + 33\right)\cdot 41^{5} + \left(33 a + 32\right)\cdot 41^{6} + \left(33 a + 4\right)\cdot 41^{7} + \left(25 a + 10\right)\cdot 41^{8} + \left(26 a + 29\right)\cdot 41^{9} +O\left(41^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,7)(6,8)$ |
| $(1,7)(2,4)(3,6)(5,8)$ |
| $(7,8)$ |
| $(1,5)$ |
| $(2,3)$ |
| $(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $4$ | $2$ | $(1,5)$ | $2$ |
| $4$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $0$ |
| $4$ | $2$ | $(1,5)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,5)$ | $-2$ |
| $4$ | $2$ | $(1,5)(2,3)(4,7)(6,8)$ | $2$ |
| $4$ | $2$ | $(1,5)(2,3)(4,6)$ | $-2$ |
| $8$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $0$ |
| $8$ | $2$ | $(1,5)(4,7)(6,8)$ | $0$ |
| $4$ | $4$ | $(1,3,5,2)(4,7,6,8)$ | $0$ |
| $4$ | $4$ | $(1,3,5,2)$ | $-2$ |
| $4$ | $4$ | $(1,5)(2,3)(4,8,6,7)$ | $2$ |
| $8$ | $4$ | $(1,8,5,7)(2,6,3,4)$ | $0$ |
| $8$ | $4$ | $(1,3,5,2)(4,7)(6,8)$ | $0$ |
| $8$ | $4$ | $(1,3,5,2)(4,6)$ | $0$ |
| $16$ | $4$ | $(1,4,2,7)(3,8,5,6)$ | $0$ |
| $16$ | $4$ | $(1,8,5,7)(2,4)(3,6)$ | $0$ |
| $16$ | $8$ | $(1,6,3,8,5,4,2,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
