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