Properties

Label 1.36.6t1.b.b
Dimension 1
Group $C_6$
Conductor $ 2^{2} \cdot 3^{2}$
Root number not computed
Frobenius-Schur indicator 0

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Basic invariants

Dimension:$1$
Group:$C_6$
Conductor:$36= 2^{2} \cdot 3^{2} $
Artin number field: Splitting field of $f= x^{6} + 6 x^{4} + 9 x^{2} + 1 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Odd
Corresponding Dirichlet character: \(\chi_{36}(31,\cdot)\)

Galois action

Roots of defining polynomial

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

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

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

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