Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{2} + 38 x + 6 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 30 a + 7 + \left(35 a + 39\right)\cdot 41 + \left(14 a + 6\right)\cdot 41^{2} + \left(9 a + 11\right)\cdot 41^{3} + \left(12 a + 26\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 30 a + \left(35 a + 37\right)\cdot 41 + \left(14 a + 21\right)\cdot 41^{2} + \left(9 a + 2\right)\cdot 41^{3} + \left(12 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 a + 23 + \left(35 a + 33\right)\cdot 41 + \left(14 a + 39\right)\cdot 41^{2} + \left(9 a + 27\right)\cdot 41^{3} + \left(12 a + 38\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 a + 8 + \left(5 a + 32\right)\cdot 41 + \left(26 a + 30\right)\cdot 41^{2} + \left(31 a + 15\right)\cdot 41^{3} + \left(28 a + 3\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 a + 15 + \left(5 a + 34\right)\cdot 41 + \left(26 a + 15\right)\cdot 41^{2} + \left(31 a + 24\right)\cdot 41^{3} + \left(28 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 11 a + 31 + \left(5 a + 28\right)\cdot 41 + \left(26 a + 7\right)\cdot 41^{2} + 31 a\cdot 41^{3} + \left(28 a + 25\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(1,5)(2,4)(3,6)$ |
| $(1,3,2)(4,5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,5)(2,4)(3,6)$ |
$-1$ |
$-1$ |
| $1$ |
$3$ |
$(1,3,2)(4,5,6)$ |
$\zeta_{3}$ |
$-\zeta_{3} - 1$ |
| $1$ |
$3$ |
$(1,2,3)(4,6,5)$ |
$-\zeta_{3} - 1$ |
$\zeta_{3}$ |
| $1$ |
$6$ |
$(1,6,2,5,3,4)$ |
$-\zeta_{3}$ |
$\zeta_{3} + 1$ |
| $1$ |
$6$ |
$(1,4,3,5,2,6)$ |
$\zeta_{3} + 1$ |
$-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
