Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(273\)\(\medspace = 3 \cdot 7 \cdot 13 \) |
| Artin field: | Galois closure of 6.0.1851523947.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{273}(191,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 31x^{4} + 84x^{3} + 873x^{2} + 810x + 729 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$:
\( x^{2} + 7x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 5 + a\cdot 11 + \left(9 a + 5\right)\cdot 11^{2} + \left(5 a + 10\right)\cdot 11^{3} + \left(3 a + 10\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 5 a + 9 + \left(5 a + 1\right)\cdot 11 + \left(a + 1\right)\cdot 11^{2} + \left(3 a + 2\right)\cdot 11^{3} + \left(2 a + 2\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 7 a + 10 + \left(9 a + 1\right)\cdot 11 + \left(a + 7\right)\cdot 11^{2} + \left(5 a + 2\right)\cdot 11^{3} + \left(7 a + 8\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 6 a + 7 + \left(5 a + 7\right)\cdot 11 + \left(9 a + 1\right)\cdot 11^{2} + \left(7 a + 2\right)\cdot 11^{3} + \left(8 a + 8\right)\cdot 11^{4} +O(11^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 3 a + 1 + \left(a + 4\right)\cdot 11 + \left(a + 7\right)\cdot 11^{2} + \left(6 a + 1\right)\cdot 11^{3} + 9\cdot 11^{4} +O(11^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 8 a + 2 + \left(9 a + 6\right)\cdot 11 + \left(9 a + 10\right)\cdot 11^{2} + \left(4 a + 2\right)\cdot 11^{3} + \left(10 a + 5\right)\cdot 11^{4} +O(11^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,5)(3,4,6)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,5,2)(3,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,4,5,3,2,6)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,6,2,3,5,4)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.