Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(171\)\(\medspace = 3^{2} \cdot 19 \) |
| Artin field: | Galois closure of 6.6.48737056617.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{171}(50,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 57x^{4} - 95x^{3} + 684x^{2} + 1596x - 152 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 19 a + 21 + 30\cdot 37 + \left(7 a + 6\right)\cdot 37^{2} + \left(22 a + 4\right)\cdot 37^{3} + \left(27 a + 10\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 28 a + 14 + \left(26 a + 24\right)\cdot 37 + \left(4 a + 28\right)\cdot 37^{2} + \left(27 a + 2\right)\cdot 37^{3} + \left(30 a + 36\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 18 a + 23 + \left(36 a + 13\right)\cdot 37 + \left(29 a + 34\right)\cdot 37^{2} + \left(14 a + 11\right)\cdot 37^{3} + \left(9 a + 24\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 a + 15 + \left(10 a + 29\right)\cdot 37 + \left(32 a + 20\right)\cdot 37^{2} + \left(9 a + 32\right)\cdot 37^{3} + \left(6 a + 20\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 28 a + \left(10 a + 36\right)\cdot 37 + \left(2 a + 10\right)\cdot 37^{2} + \left(32 a + 22\right)\cdot 37^{3} + \left(33 a + 13\right)\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 a + 1 + \left(26 a + 14\right)\cdot 37 + \left(34 a + 9\right)\cdot 37^{2} + 4 a\cdot 37^{3} + \left(3 a + 6\right)\cdot 37^{4} +O(37^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-1$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,6,3,4,5)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,5,4,3,6,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.