Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(456\)\(\medspace = 2^{3} \cdot 3 \cdot 19 \) |
| Artin field: | Galois closure of 6.6.1801557504.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{456}(83,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} - 29x^{4} + 50x^{3} + 118x^{2} + 24x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$:
\( x^{2} + 6x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 1 + 2\cdot 7 + 6\cdot 7^{2} + 5\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 6 a + 4 + \left(6 a + 5\right)\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + 6 a\cdot 7^{3} + \left(6 a + 6\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 6 a + \left(6 a + 6\right)\cdot 7 + 6 a\cdot 7^{2} + 6 a\cdot 7^{3} + \left(6 a + 5\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 4 }$ |
$=$ |
\( a + 6 + 6\cdot 7 + 5\cdot 7^{4} +O(7^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 6 a + 2 + \left(6 a + 1\right)\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + \left(6 a + 5\right)\cdot 7^{3} + \left(6 a + 2\right)\cdot 7^{4} +O(7^{5})\)
| $r_{ 6 }$ |
$=$ |
\( a + 3 + 6\cdot 7 + 6\cdot 7^{2} + 6\cdot 7^{4} +O(7^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,2,4,5,6,3)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,5,4,2)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.