Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(572\)\(\medspace = 2^{2} \cdot 11 \cdot 13 \) |
| Artin field: | Galois closure of 6.6.31628222912.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{572}(43,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 143x^{4} + 3146x^{2} - 17303 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 83 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 83 }$:
\( x^{2} + 82x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 46 a + 60 + \left(4 a + 20\right)\cdot 83 + \left(27 a + 30\right)\cdot 83^{2} + \left(49 a + 30\right)\cdot 83^{3} + \left(5 a + 63\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 36 a + 65 + \left(53 a + 32\right)\cdot 83 + \left(37 a + 49\right)\cdot 83^{2} + \left(28 a + 4\right)\cdot 83^{3} + \left(44 a + 75\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a + 81 + \left(37 a + 24\right)\cdot 83 + \left(68 a + 67\right)\cdot 83^{2} + \left(73 a + 38\right)\cdot 83^{3} + \left(15 a + 70\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 37 a + 23 + \left(78 a + 62\right)\cdot 83 + \left(55 a + 52\right)\cdot 83^{2} + \left(33 a + 52\right)\cdot 83^{3} + \left(77 a + 19\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 47 a + 18 + \left(29 a + 50\right)\cdot 83 + \left(45 a + 33\right)\cdot 83^{2} + \left(54 a + 78\right)\cdot 83^{3} + \left(38 a + 7\right)\cdot 83^{4} +O(83^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 79 a + 2 + \left(45 a + 58\right)\cdot 83 + \left(14 a + 15\right)\cdot 83^{2} + \left(9 a + 44\right)\cdot 83^{3} + \left(67 a + 12\right)\cdot 83^{4} +O(83^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,2)(3,5,4)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,2,6)(3,4,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,3,2,4,6,5)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,5,6,4,2,3)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.