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