Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(1007\)\(\medspace = 19 \cdot 53 \) |
| Artin field: | Galois closure of 6.0.368634190823.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{1007}(635,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{5} + 249x^{4} + 255x^{3} + 6421x^{2} + 6417x + 51877 \)
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The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{2} + 33x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a + 10 + \left(35 a + 19\right)\cdot 37 + \left(23 a + 4\right)\cdot 37^{2} + \left(12 a + 19\right)\cdot 37^{3} + \left(25 a + 22\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 30 a + 18 + \left(29 a + 1\right)\cdot 37 + \left(6 a + 6\right)\cdot 37^{2} + \left(25 a + 13\right)\cdot 37^{3} + \left(33 a + 18\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a + 27 + \left(7 a + 16\right)\cdot 37 + \left(30 a + 3\right)\cdot 37^{2} + \left(11 a + 33\right)\cdot 37^{3} + \left(3 a + 16\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a + 34 + \left(a + 5\right)\cdot 37 + \left(13 a + 28\right)\cdot 37^{2} + \left(24 a + 8\right)\cdot 37^{3} + 11 a\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 20 + \left(21 a + 12\right)\cdot 37 + \left(3 a + 19\right)\cdot 37^{2} + \left(2 a + 34\right)\cdot 37^{3} + 8\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 32 a + 3 + \left(15 a + 18\right)\cdot 37 + \left(33 a + 12\right)\cdot 37^{2} + \left(34 a + 2\right)\cdot 37^{3} + \left(36 a + 7\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,2,6)(3,5,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,6,2)(3,4,5)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,3,6,4,2,5)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,5,2,4,6,3)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.