Properties

Label 1.2e2_3e2.6t1.1c1
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} - 3 $ over $\Q$
Size of Galois orbit: 2
Smallest containing permutation representation: $C_6$
Parity: Even
Corresponding Dirichlet character: \(\chi_{36}(23,\cdot)\)

Galois action

Roots of defining polynomial

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 $
Roots: \[ \begin{aligned} r_{ 1 } &= 14 a + 10 + \left(5 a + 12\right)\cdot 17 + \left(11 a + 5\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 2 } &= 12 a + 11 + \left(3 a + 12\right)\cdot 17 + \left(9 a + 5\right)\cdot 17^{2} + \left(2 a + 3\right)\cdot 17^{3} + \left(6 a + 15\right)\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 3 } &= 2 a + 16 + \left(2 a + 16\right)\cdot 17 + \left(2 a + 16\right)\cdot 17^{2} + \left(15 a + 1\right)\cdot 17^{3} + \left(5 a + 13\right)\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 4 } &= 3 a + 7 + \left(11 a + 4\right)\cdot 17 + \left(5 a + 11\right)\cdot 17^{2} + \left(16 a + 11\right)\cdot 17^{3} + \left(4 a + 5\right)\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 5 } &= 5 a + 6 + \left(13 a + 4\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(14 a + 13\right)\cdot 17^{3} + \left(10 a + 1\right)\cdot 17^{4} +O\left(17^{ 5 }\right) \\ r_{ 6 } &= 15 a + 1 + 14 a\cdot 17 + 14 a\cdot 17^{2} + \left(a + 15\right)\cdot 17^{3} + \left(11 a + 3\right)\cdot 17^{4} +O\left(17^{ 5 }\right) \\ \end{aligned}\]

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