Properties

Label 1.45.6t1.b.b
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: 6.0.2460375.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: odd
Dirichlet character: \(\chi_{45}(14,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} + 6 x^{4} - 4 x^{3} + 9 x^{2} - 12 x + 19\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \(x^{2} + 33 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a + \left(23 a + 3\right)\cdot 37 + \left(28 a + 26\right)\cdot 37^{2} + \left(13 a + 35\right)\cdot 37^{3} + \left(20 a + 35\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 29 a + 32 + \left(13 a + 13\right)\cdot 37 + \left(8 a + 6\right)\cdot 37^{2} + \left(23 a + 25\right)\cdot 37^{3} + \left(16 a + 29\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 32 a + 24 + \left(30 a + 27\right)\cdot 37 + \left(18 a + 17\right)\cdot 37^{2} + \left(10 a + 16\right)\cdot 37^{3} + \left(23 a + 34\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 1 + \left(17 a + 15\right)\cdot 37 + \left(10 a + 5\right)\cdot 37^{2} + \left(24 a + 9\right)\cdot 37^{3} + \left(6 a + 1\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + 4 + \left(6 a + 8\right)\cdot 37 + \left(18 a + 25\right)\cdot 37^{2} + \left(26 a + 2\right)\cdot 37^{3} + \left(13 a + 6\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 34 a + 13 + \left(19 a + 6\right)\cdot 37 + \left(26 a + 30\right)\cdot 37^{2} + \left(12 a + 21\right)\cdot 37^{3} + \left(30 a + 3\right)\cdot 37^{4} +O(37^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.