Properties

Label 1.45.6t1.a.a
Dimension $1$
Group $C_6$
Conductor $45$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(45\)\(\medspace = 3^{2} \cdot 5 \)
Artin field: 6.6.820125.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{45}(34,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 11 a + 5 + \left(3 a + 8\right)\cdot 17 + \left(a + 16\right)\cdot 17^{2} + \left(14 a + 15\right)\cdot 17^{3} + \left(14 a + 15\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 13 a + 12 + \left(15 a + 4\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(11 a + 5\right)\cdot 17^{3} + \left(12 a + 10\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 4 a + 8 + \left(a + 7\right)\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + \left(5 a + 9\right)\cdot 17^{3} + \left(4 a + 11\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 15 a + 6 + \left(4 a + 11\right)\cdot 17 + \left(10 a + 8\right)\cdot 17^{2} + \left(2 a + 16\right)\cdot 17^{3} + \left(2 a + 6\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 2 a + 4 + \left(12 a + 1\right)\cdot 17 + \left(6 a + 14\right)\cdot 17^{2} + \left(14 a + 8\right)\cdot 17^{3} + \left(14 a + 6\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 6 a + 16 + 13 a\cdot 17 + \left(15 a + 14\right)\cdot 17^{2} + \left(2 a + 11\right)\cdot 17^{3} + \left(2 a + 16\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$1$
$1$$2$$(1,6)(2,3)(4,5)$$-1$
$1$$3$$(1,5,3)(2,6,4)$$\zeta_{3}$
$1$$3$$(1,3,5)(2,4,6)$$-\zeta_{3} - 1$
$1$$6$$(1,2,5,6,3,4)$$\zeta_{3} + 1$
$1$$6$$(1,4,3,6,5,2)$$-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.