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