Properties

Label 1.72.6t1.a.a
Dimension 1
Group $C_6$
Conductor $ 2^{3} \cdot 3^{2}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$72= 2^{3} \cdot 3^{2} $
Artin number field: Splitting field of 6.0.3359232.1 defined by $f= x^{6} + 12 x^{4} + 36 x^{2} + 8 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{72}(67,\cdot)\)
Projective image: $C_1$
Projective field: \(\Q\)

Galois action

Roots of defining polynomial

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 $
Roots:
$r_{ 1 }$ $=$ $ 10 a + 17 + \left(27 a + 24\right)\cdot 37 + \left(22 a + 23\right)\cdot 37^{2} + \left(5 a + 18\right)\cdot 37^{3} + \left(26 a + 24\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 2 }$ $=$ $ 20 a + 34 + \left(12 a + 21\right)\cdot 37 + \left(19 a + 4\right)\cdot 37^{2} + \left(22 a + 20\right)\cdot 37^{3} + \left(30 a + 5\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 3 }$ $=$ $ 7 a + 23 + \left(34 a + 27\right)\cdot 37 + \left(31 a + 8\right)\cdot 37^{2} + \left(8 a + 35\right)\cdot 37^{3} + \left(17 a + 6\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 4 }$ $=$ $ 27 a + 20 + \left(9 a + 12\right)\cdot 37 + \left(14 a + 13\right)\cdot 37^{2} + \left(31 a + 18\right)\cdot 37^{3} + \left(10 a + 12\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 5 }$ $=$ $ 17 a + 3 + \left(24 a + 15\right)\cdot 37 + \left(17 a + 32\right)\cdot 37^{2} + \left(14 a + 16\right)\cdot 37^{3} + \left(6 a + 31\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$
$r_{ 6 }$ $=$ $ 30 a + 14 + \left(2 a + 9\right)\cdot 37 + \left(5 a + 28\right)\cdot 37^{2} + \left(28 a + 1\right)\cdot 37^{3} + \left(19 a + 30\right)\cdot 37^{4} +O\left(37^{ 5 }\right)$

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

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

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$$3$$(1,2,3)(4,5,6)$$\zeta_{3}$
$1$$3$$(1,3,2)(4,6,5)$$-\zeta_{3} - 1$
$1$$6$$(1,5,3,4,2,6)$$-\zeta_{3}$
$1$$6$$(1,6,2,4,3,5)$$\zeta_{3} + 1$
The blue line marks the conjugacy class containing complex conjugation.