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:
| $r_{ 1 }$ |
$=$ |
$ 38 a + 8 + \left(25 a + 13\right)\cdot 41 + \left(22 a + 37\right)\cdot 41^{2} + \left(24 a + 20\right)\cdot 41^{3} + \left(18 a + 36\right)\cdot 41^{4} + \left(38 a + 40\right)\cdot 41^{5} + \left(26 a + 6\right)\cdot 41^{6} + \left(12 a + 38\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 a + 40 + \left(15 a + 11\right)\cdot 41 + \left(18 a + 38\right)\cdot 41^{2} + \left(16 a + 30\right)\cdot 41^{3} + \left(22 a + 26\right)\cdot 41^{4} + \left(2 a + 14\right)\cdot 41^{5} + \left(14 a + 8\right)\cdot 41^{6} + \left(28 a + 8\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 a + 34 + \left(15 a + 27\right)\cdot 41 + \left(18 a + 3\right)\cdot 41^{2} + \left(16 a + 20\right)\cdot 41^{3} + \left(22 a + 4\right)\cdot 41^{4} + 2 a\cdot 41^{5} + \left(14 a + 34\right)\cdot 41^{6} + \left(28 a + 2\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 25\cdot 41 + 33\cdot 41^{2} + 17\cdot 41^{3} + 12\cdot 41^{4} + 35\cdot 41^{5} + 3\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 15\cdot 41 + 7\cdot 41^{2} + 23\cdot 41^{3} + 28\cdot 41^{4} + 5\cdot 41^{5} + 40\cdot 41^{6} + 37\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 a + 2 + \left(25 a + 29\right)\cdot 41 + \left(22 a + 2\right)\cdot 41^{2} + \left(24 a + 10\right)\cdot 41^{3} + \left(18 a + 14\right)\cdot 41^{4} + \left(38 a + 26\right)\cdot 41^{5} + \left(26 a + 32\right)\cdot 41^{6} + \left(12 a + 32\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,4,2)(3,5,6)$ |
| $(4,5)$ |
| $(2,4)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,3)(2,6)(4,5)$ |
$-3$ |
| $3$ |
$2$ |
$(1,3)$ |
$1$ |
| $3$ |
$2$ |
$(1,3)(4,5)$ |
$-1$ |
| $6$ |
$2$ |
$(2,4)(5,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,3)(2,4)(5,6)$ |
$1$ |
| $8$ |
$3$ |
$(1,4,2)(3,5,6)$ |
$0$ |
| $6$ |
$4$ |
$(1,5,3,4)$ |
$-1$ |
| $6$ |
$4$ |
$(1,6,3,2)(4,5)$ |
$1$ |
| $8$ |
$6$ |
$(1,5,6,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
