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:
| $r_{ 1 }$ |
$=$ |
$ 24 a + 21 + \left(16 a + 13\right)\cdot 41 + \left(36 a + 20\right)\cdot 41^{2} + \left(18 a + 12\right)\cdot 41^{3} + \left(30 a + 5\right)\cdot 41^{4} + \left(37 a + 37\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 a + 4 + \left(14 a + 14\right)\cdot 41 + \left(8 a + 7\right)\cdot 41^{2} + \left(23 a + 38\right)\cdot 41^{3} + \left(11 a + 35\right)\cdot 41^{4} + \left(40 a + 13\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 36\cdot 41 + 17\cdot 41^{2} + 21\cdot 41^{3} + 36\cdot 41^{4} + 32\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 3 + 23\cdot 41 + 28\cdot 41^{2} + 2\cdot 41^{4} + 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 a + 11 + \left(24 a + 39\right)\cdot 41 + \left(4 a + 30\right)\cdot 41^{2} + \left(22 a + 32\right)\cdot 41^{3} + \left(10 a + 36\right)\cdot 41^{4} + \left(3 a + 37\right)\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 a + 23 + \left(26 a + 37\right)\cdot 41 + \left(32 a + 17\right)\cdot 41^{2} + \left(17 a + 17\right)\cdot 41^{3} + \left(29 a + 6\right)\cdot 41^{4} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,2,3,6,5,4)$ |
| $(2,4)(3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $3$ | $2$ | $(2,4)(3,5)$ | $0$ |
| $3$ | $2$ | $(1,2)(3,4)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,3,5)(2,6,4)$ | $-1$ |
| $2$ | $6$ | $(1,2,3,6,5,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
