Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(72\)\(\medspace = 2^{3} \cdot 3^{2}\) |
| Artin field: | Galois closure of 6.6.3359232.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{72}(13,\cdot)\) |
| 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 $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$:
\( x^{2} + 18x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 a + 12 + \left(11 a + 10\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + \left(12 a + 18\right)\cdot 19^{3} + \left(14 a + 17\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 18 a + 10 + 8\cdot 19 + \left(5 a + 7\right)\cdot 19^{2} + \left(a + 11\right)\cdot 19^{3} + \left(16 a + 11\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 15 a + 2 + \left(10 a + 2\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(11 a + 7\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 5 a + 7 + \left(7 a + 8\right)\cdot 19 + \left(7 a + 9\right)\cdot 19^{2} + 6 a\cdot 19^{3} + \left(4 a + 1\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 5 }$ |
$=$ |
\( a + 9 + \left(18 a + 10\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{2} + \left(17 a + 7\right)\cdot 19^{3} + \left(2 a + 7\right)\cdot 19^{4} +O(19^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 4 a + 17 + \left(8 a + 16\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(7 a + 11\right)\cdot 19^{3} + \left(a + 12\right)\cdot 19^{4} +O(19^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,6)(2,3,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,5)(2,4,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,5,4,6,2)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,2,6,4,5,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.