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