Properties

Label 2.216.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $216$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(216\)\(\medspace = 2^{3} \cdot 3^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.0.139968.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.24.2t1.b.a
Projective image: $S_3$
Projective stem field: 3.1.216.1

Defining polynomial

$f(x)$$=$\(x^{6} - 3 x^{5} + 3 x^{3} + 6 x^{2} - 9 x + 3\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \(x^{2} + 12 x + 2\)  Toggle raw display

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.