Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(45\)\(\medspace = 3^{2} \cdot 5 \) |
| Artin field: | Galois closure of 6.0.2460375.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{45}(29,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 6x^{4} - 4x^{3} + 9x^{2} - 12x + 19 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a + \left(23 a + 3\right)\cdot 37 + \left(28 a + 26\right)\cdot 37^{2} + \left(13 a + 35\right)\cdot 37^{3} + \left(20 a + 35\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 29 a + 32 + \left(13 a + 13\right)\cdot 37 + \left(8 a + 6\right)\cdot 37^{2} + \left(23 a + 25\right)\cdot 37^{3} + \left(16 a + 29\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 32 a + 24 + \left(30 a + 27\right)\cdot 37 + \left(18 a + 17\right)\cdot 37^{2} + \left(10 a + 16\right)\cdot 37^{3} + \left(23 a + 34\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 3 a + 1 + \left(17 a + 15\right)\cdot 37 + \left(10 a + 5\right)\cdot 37^{2} + \left(24 a + 9\right)\cdot 37^{3} + \left(6 a + 1\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 5 a + 4 + \left(6 a + 8\right)\cdot 37 + \left(18 a + 25\right)\cdot 37^{2} + \left(26 a + 2\right)\cdot 37^{3} + \left(13 a + 6\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 34 a + 13 + \left(19 a + 6\right)\cdot 37 + \left(26 a + 30\right)\cdot 37^{2} + \left(12 a + 21\right)\cdot 37^{3} + \left(30 a + 3\right)\cdot 37^{4} +O(37^{5})\)
| |
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,3,6)(2,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,3)(2,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,6,2,3,4)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,4,3,2,6,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.