Basic invariants
Dimension: | $1$ |
Group: | $C_6$ |
Conductor: | \(9\)\(\medspace = 3^{2} \) |
Artin field: | Galois closure of \(\Q(\zeta_{9})\) |
Galois orbit size: | $2$ |
Smallest permutation container: | $C_6$ |
Parity: | odd |
Dirichlet character: | \(\chi_{9}(5,\cdot)\) |
Projective image: | $C_1$ |
Projective field: | Galois closure of \(\Q\) |
Defining polynomial
$f(x)$ | $=$ |
\( x^{6} - x^{3} + 1 \)
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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:
$r_{ 1 }$ | $=$ |
\( 11 a + 5 + \left(8 a + 14\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + \left(8 a + 7\right)\cdot 17^{3} + \left(a + 11\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 2 }$ | $=$ |
\( 4 a + 3 + \left(6 a + 5\right)\cdot 17 + \left(16 a + 6\right)\cdot 17^{2} + \left(13 a + 5\right)\cdot 17^{3} + \left(3 a + 3\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 3 }$ | $=$ |
\( 10 a + 5 + \left(14 a + 12\right)\cdot 17 + \left(2 a + 4\right)\cdot 17^{2} + \left(5 a + 6\right)\cdot 17^{3} + \left(2 a + 12\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 4 }$ | $=$ |
\( 7 a + 15 + \left(2 a + 16\right)\cdot 17 + \left(14 a + 9\right)\cdot 17^{2} + \left(11 a + 8\right)\cdot 17^{3} + \left(14 a + 9\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 5 }$ | $=$ |
\( 13 a + 7 + \left(10 a + 7\right)\cdot 17 + 16\cdot 17^{2} + \left(3 a + 2\right)\cdot 17^{3} + \left(13 a + 10\right)\cdot 17^{4} +O(17^{5})\)
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$r_{ 6 }$ | $=$ |
\( 6 a + 16 + \left(8 a + 11\right)\cdot 17 + 3 a\cdot 17^{2} + \left(8 a + 3\right)\cdot 17^{3} + \left(15 a + 4\right)\cdot 17^{4} +O(17^{5})\)
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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 | Complex conjugation |
$1$ | $1$ | $()$ | $1$ | |
$1$ | $2$ | $(1,6)(2,5)(3,4)$ | $-1$ | ✓ |
$1$ | $3$ | $(1,3,5)(2,6,4)$ | $\zeta_{3}$ | |
$1$ | $3$ | $(1,5,3)(2,4,6)$ | $-\zeta_{3} - 1$ | |
$1$ | $6$ | $(1,2,3,6,5,4)$ | $\zeta_{3} + 1$ | |
$1$ | $6$ | $(1,4,5,6,3,2)$ | $-\zeta_{3}$ |