Properties

Label 1.9.6t1.a.a
Dimension $1$
Group $C_6$
Conductor $9$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(9\)\(\medspace = 3^{2}\)
Artin number 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: \(\Q\)

Defining polynomial

$f(x)$$=$$ x^{6} - x^{3} + 1 $.

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} + 16 x + 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\left(17^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 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\left(17^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 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\left(17^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 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\left(17^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 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\left(17^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 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\left(17^{ 5 }\right)$

Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

Cycle notation
$(1,2,3,6,5,4)$
$(1,6)(2,5)(3,4)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$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}$

The blue line marks the conjugacy class containing complex conjugation.