Basic invariants
| Dimension: | $1$ |
| Group: | $C_6$ |
| Conductor: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
| Artin field: | Galois closure of 6.6.144027072.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_6$ |
| Parity: | even |
| Dirichlet character: | \(\chi_{252}(139,\cdot)\) |
| Projective image: | $C_1$ |
| Projective field: | Galois closure of \(\Q\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 27x^{4} - 2x^{3} + 156x^{2} - 36x - 111 \)
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The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$:
\( x^{2} + 16x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 a + 1 + \left(8 a + 10\right)\cdot 17 + \left(8 a + 14\right)\cdot 17^{2} + \left(14 a + 11\right)\cdot 17^{3} + \left(16 a + 3\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a + 6 + \left(8 a + 11\right)\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(2 a + 13\right)\cdot 17^{3} + 16\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 a + 12 + \left(8 a + 14\right)\cdot 17 + \left(8 a + 5\right)\cdot 17^{2} + \left(2 a + 11\right)\cdot 17^{3} + 14\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 4 a + 8 + \left(8 a + 10\right)\cdot 17 + \left(8 a + 5\right)\cdot 17^{2} + \left(14 a + 5\right)\cdot 17^{3} + \left(16 a + 12\right)\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 13 a + 5 + \left(8 a + 14\right)\cdot 17 + \left(8 a + 14\right)\cdot 17^{2} + 2 a\cdot 17^{3} + 6\cdot 17^{4} +O(17^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 4 a + 2 + \left(8 a + 7\right)\cdot 17 + \left(8 a + 13\right)\cdot 17^{2} + \left(14 a + 7\right)\cdot 17^{3} + \left(16 a + 14\right)\cdot 17^{4} +O(17^{5})\)
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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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,5)(2,6)(3,4)$ | $-1$ |
| $1$ | $3$ | $(1,4,6)(2,5,3)$ | $\zeta_{3}$ |
| $1$ | $3$ | $(1,6,4)(2,3,5)$ | $-\zeta_{3} - 1$ |
| $1$ | $6$ | $(1,2,4,5,6,3)$ | $\zeta_{3} + 1$ |
| $1$ | $6$ | $(1,3,6,5,4,2)$ | $-\zeta_{3}$ |
The blue line marks the conjugacy class containing complex conjugation.