Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Artin field: | Galois closure of 6.0.45001899.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | odd |
| Dirichlet character: | \(\chi_{171}(94,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 3x^{5} + 12x^{4} - 17x^{3} + 87x^{2} - 114x + 323 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 20 a + 21 + \left(11 a + 25\right)\cdot 37 + \left(26 a + 31\right)\cdot 37^{2} + \left(3 a + 18\right)\cdot 37^{3} + \left(31 a + 3\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 a + 31 + \left(25 a + 32\right)\cdot 37 + \left(10 a + 3\right)\cdot 37^{2} + \left(33 a + 12\right)\cdot 37^{3} + \left(5 a + 1\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 a + 2 + \left(11 a + 21\right)\cdot 37 + \left(26 a + 28\right)\cdot 37^{2} + \left(3 a + 11\right)\cdot 37^{3} + \left(31 a + 27\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 27 + \left(25 a + 14\right)\cdot 37 + \left(10 a + 14\right)\cdot 37^{2} + \left(33 a + 7\right)\cdot 37^{3} + \left(5 a + 13\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 20 a + 25 + \left(11 a + 6\right)\cdot 37 + \left(26 a + 21\right)\cdot 37^{2} + \left(3 a + 23\right)\cdot 37^{3} + \left(31 a + 28\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 17 a + 8 + \left(25 a + 10\right)\cdot 37 + \left(10 a + 11\right)\cdot 37^{2} + 33 a\cdot 37^{3} + 5 a\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,5)(3,6)$ | $-1$ |
| $1$ | $3$ | $(1,5,3)(2,6,4)$ | $-\zeta_{3} - 1$ |
| $1$ | $3$ | $(1,3,5)(2,4,6)$ | $\zeta_{3}$ |
| $1$ | $6$ | $(1,6,5,4,3,2)$ | $-\zeta_{3}$ |
| $1$ | $6$ | $(1,2,3,4,5,6)$ | $\zeta_{3} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.