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