Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(72\)\(\medspace = 2^{3} \cdot 3^{2}\) |
| Artin field: | Galois closure of 6.0.3359232.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{72}(67,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} + 12x^{4} + 36x^{2} + 8 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 10 a + 17 + \left(27 a + 24\right)\cdot 37 + \left(22 a + 23\right)\cdot 37^{2} + \left(5 a + 18\right)\cdot 37^{3} + \left(26 a + 24\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 20 a + 34 + \left(12 a + 21\right)\cdot 37 + \left(19 a + 4\right)\cdot 37^{2} + \left(22 a + 20\right)\cdot 37^{3} + \left(30 a + 5\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 7 a + 23 + \left(34 a + 27\right)\cdot 37 + \left(31 a + 8\right)\cdot 37^{2} + \left(8 a + 35\right)\cdot 37^{3} + \left(17 a + 6\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 27 a + 20 + \left(9 a + 12\right)\cdot 37 + \left(14 a + 13\right)\cdot 37^{2} + \left(31 a + 18\right)\cdot 37^{3} + \left(10 a + 12\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 17 a + 3 + \left(24 a + 15\right)\cdot 37 + \left(17 a + 32\right)\cdot 37^{2} + \left(14 a + 16\right)\cdot 37^{3} + \left(6 a + 31\right)\cdot 37^{4} +O(37^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 30 a + 14 + \left(2 a + 9\right)\cdot 37 + \left(5 a + 28\right)\cdot 37^{2} + \left(28 a + 1\right)\cdot 37^{3} + \left(19 a + 30\right)\cdot 37^{4} +O(37^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,5,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,6,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,5,3,4,2,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,2,4,3,5)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.