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:
| $r_{ 1 }$ |
$=$ |
$ 10 a + 33 + \left(20 a + 1\right)\cdot 41 + \left(25 a + 8\right)\cdot 41^{2} + 35 a\cdot 41^{3} + \left(19 a + 33\right)\cdot 41^{4} + \left(15 a + 26\right)\cdot 41^{5} + \left(18 a + 4\right)\cdot 41^{6} + 37\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + 17 + \left(3 a + 26\right)\cdot 41 + \left(14 a + 20\right)\cdot 41^{2} + \left(22 a + 1\right)\cdot 41^{3} + \left(37 a + 16\right)\cdot 41^{4} + \left(24 a + 10\right)\cdot 41^{5} + \left(38 a + 40\right)\cdot 41^{6} + \left(20 a + 38\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 34 + 38\cdot 41 + 33\cdot 41^{2} + 29\cdot 41^{3} + 29\cdot 41^{4} + 40\cdot 41^{5} + 17\cdot 41^{6} + 19\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 36\cdot 41 + 18\cdot 41^{2} + 37\cdot 41^{3} + 3\cdot 41^{4} + 26\cdot 41^{5} + 7\cdot 41^{6} + 26\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 a + 22 + \left(20 a + 11\right)\cdot 41 + \left(15 a + 23\right)\cdot 41^{2} + \left(5 a + 40\right)\cdot 41^{3} + \left(21 a + 15\right)\cdot 41^{4} + \left(25 a + 12\right)\cdot 41^{5} + \left(22 a + 3\right)\cdot 41^{6} + \left(40 a + 20\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 25 + \left(37 a + 7\right)\cdot 41 + \left(26 a + 18\right)\cdot 41^{2} + \left(18 a + 13\right)\cdot 41^{3} + \left(3 a + 24\right)\cdot 41^{4} + \left(16 a + 6\right)\cdot 41^{5} + \left(2 a + 8\right)\cdot 41^{6} + \left(20 a + 22\right)\cdot 41^{7} +O\left(41^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,3)(4,6)$ |
| $(1,2,3)(4,6,5)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $1$ |
$2$ |
$(1,6)(2,5)(3,4)$ |
$-3$ |
| $3$ |
$2$ |
$(1,6)(3,4)$ |
$-1$ |
| $3$ |
$2$ |
$(1,6)$ |
$1$ |
| $6$ |
$2$ |
$(1,3)(4,6)$ |
$-1$ |
| $6$ |
$2$ |
$(1,6)(2,3)(4,5)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$0$ |
| $6$ |
$4$ |
$(1,4,6,3)$ |
$-1$ |
| $6$ |
$4$ |
$(1,4,6,3)(2,5)$ |
$1$ |
| $8$ |
$6$ |
$(1,4,5,6,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
