Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(111\)\(\medspace = 3 \cdot 37 \) |
| Artin field: | Galois closure of 6.0.50602347.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{111}(26,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 13x^{4} + 34x^{3} + 133x^{2} + 132x + 121 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a + 5 + 13\cdot 29 + \left(5 a + 4\right)\cdot 29^{2} + \left(27 a + 2\right)\cdot 29^{3} + \left(27 a + 27\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 18 a + 24 + \left(24 a + 24\right)\cdot 29 + 28\cdot 29^{2} + \left(19 a + 21\right)\cdot 29^{3} + \left(27 a + 26\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 13 a + 27 + \left(28 a + 28\right)\cdot 29 + \left(23 a + 28\right)\cdot 29^{2} + \left(a + 16\right)\cdot 29^{3} + \left(a + 23\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 17 a + 18 + \left(7 a + 21\right)\cdot 29 + 10 a\cdot 29^{2} + \left(10 a + 2\right)\cdot 29^{3} + 15 a\cdot 29^{4} +O(29^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 11 a + 27 + \left(4 a + 13\right)\cdot 29 + \left(28 a + 8\right)\cdot 29^{2} + 9 a\cdot 29^{3} + \left(a + 1\right)\cdot 29^{4} +O(29^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 12 a + 16 + \left(21 a + 13\right)\cdot 29 + \left(18 a + 15\right)\cdot 29^{2} + \left(18 a + 14\right)\cdot 29^{3} + \left(13 a + 8\right)\cdot 29^{4} +O(29^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,4)(2,6,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,4,5)(2,3,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,3,4,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,4,3,5,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.