Properties

Label 1.72.6t1.d.b
Dimension $1$
Group $C_6$
Conductor $72$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $1$
Group: $C_6$
Conductor: \(72\)\(\medspace = 2^{3} \cdot 3^{2}\)
Artin field: 6.6.3359232.1
Galois orbit size: $2$
Smallest permutation container: $C_6$
Parity: even
Dirichlet character: \(\chi_{72}(61,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Defining polynomial

$f(x)$$=$\(x^{6} - 12 x^{4} + 36 x^{2} - 8\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \(x^{2} + 18 x + 2\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 14 a + 12 + \left(11 a + 10\right)\cdot 19 + \left(11 a + 9\right)\cdot 19^{2} + \left(12 a + 18\right)\cdot 19^{3} + \left(14 a + 17\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 18 a + 10 + 8\cdot 19 + \left(5 a + 7\right)\cdot 19^{2} + \left(a + 11\right)\cdot 19^{3} + \left(16 a + 11\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 15 a + 2 + \left(10 a + 2\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(11 a + 7\right)\cdot 19^{3} + \left(17 a + 6\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 5 a + 7 + \left(7 a + 8\right)\cdot 19 + \left(7 a + 9\right)\cdot 19^{2} + 6 a\cdot 19^{3} + \left(4 a + 1\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( a + 9 + \left(18 a + 10\right)\cdot 19 + \left(13 a + 11\right)\cdot 19^{2} + \left(17 a + 7\right)\cdot 19^{3} + \left(2 a + 7\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 17 + \left(8 a + 16\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(7 a + 11\right)\cdot 19^{3} + \left(a + 12\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.