Properties

Label 2.648.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $648$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(648\)\(\medspace = 2^{3} \cdot 3^{4}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.1259712.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $S_3$
Projective stem field: 3.1.648.1

Defining polynomial

$f(x)$$=$\(x^{6} - 3 x^{4} + 9 x^{2} - 18 x + 12\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \(x^{2} + 6 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 + 2\cdot 7 + 6\cdot 7^{2} + 6\cdot 7^{3} + 6\cdot 7^{4} + 6\cdot 7^{5} + 6\cdot 7^{6} + 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 2 + 5\cdot 7 + 4\cdot 7^{2} + 6\cdot 7^{3} + 2\cdot 7^{4} + 3\cdot 7^{5} + 7^{6} + 4\cdot 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 1 + 3\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + \left(5 a + 1\right)\cdot 7^{4} + \left(5 a + 3\right)\cdot 7^{5} + \left(3 a + 4\right)\cdot 7^{6} +O(7^{8})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 6 a + 3 + \left(a + 6\right)\cdot 7 + \left(5 a + 2\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(a + 6\right)\cdot 7^{4} + 5\cdot 7^{5} + \left(a + 5\right)\cdot 7^{6} + \left(6 a + 5\right)\cdot 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + 3 + \left(6 a + 1\right)\cdot 7 + \left(4 a + 1\right)\cdot 7^{2} + \left(6 a + 6\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + \left(a + 3\right)\cdot 7^{5} + \left(3 a + 2\right)\cdot 7^{6} + \left(6 a + 4\right)\cdot 7^{7} +O(7^{8})\)  Toggle raw display
$r_{ 6 }$ $=$ \( a + 2 + \left(5 a + 2\right)\cdot 7 + \left(a + 6\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(5 a + 4\right)\cdot 7^{4} + \left(6 a + 4\right)\cdot 7^{5} + \left(5 a + 6\right)\cdot 7^{6} + 3\cdot 7^{7} +O(7^{8})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,2)(3,4)(5,6)$$-2$
$3$$2$$(1,3)(2,4)$$0$
$3$$2$$(1,4)(2,3)(5,6)$$0$
$2$$3$$(1,5,3)(2,6,4)$$-1$
$2$$6$$(1,6,3,2,5,4)$$1$

The blue line marks the conjugacy class containing complex conjugation.