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:
| $r_{ 1 }$ |
$=$ |
$ 7 a + 36 + \left(19 a + 23\right)\cdot 41 + \left(36 a + 1\right)\cdot 41^{2} + \left(26 a + 38\right)\cdot 41^{3} + \left(17 a + 6\right)\cdot 41^{4} + \left(28 a + 21\right)\cdot 41^{5} + \left(17 a + 18\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 19 + 6\cdot 41 + 14\cdot 41^{2} + 22\cdot 41^{3} + 26\cdot 41^{4} + 2\cdot 41^{5} + 7\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 a + 10 + \left(25 a + 16\right)\cdot 41 + 13\cdot 41^{2} + \left(38 a + 32\right)\cdot 41^{3} + \left(19 a + 35\right)\cdot 41^{4} + \left(31 a + 2\right)\cdot 41^{5} + \left(13 a + 17\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 7\cdot 41 + 12\cdot 41^{2} + 7\cdot 41^{3} + 4\cdot 41^{4} + 12\cdot 41^{5} + 10\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 34 a + 16 + \left(21 a + 33\right)\cdot 41 + \left(4 a + 9\right)\cdot 41^{2} + 14 a\cdot 41^{3} + \left(23 a + 33\right)\cdot 41^{4} + \left(12 a + 6\right)\cdot 41^{5} + \left(23 a + 2\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 a + 17 + \left(15 a + 35\right)\cdot 41 + \left(40 a + 30\right)\cdot 41^{2} + \left(2 a + 22\right)\cdot 41^{3} + \left(21 a + 16\right)\cdot 41^{4} + \left(9 a + 36\right)\cdot 41^{5} + \left(27 a + 26\right)\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,5)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-3$ |
| $3$ |
$2$ |
$(1,5)$ |
$1$ |
| $3$ |
$2$ |
$(1,5)(2,4)$ |
$-1$ |
| $4$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $4$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$0$ |
| $4$ |
$6$ |
$(1,4,6,5,2,3)$ |
$0$ |
| $4$ |
$6$ |
$(1,3,2,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
