Properties

Label 2.1728.6t3.c.a
Dimension $2$
Group $D_{6}$
Conductor $1728$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(1728\)\(\medspace = 2^{6} \cdot 3^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.1492992.4
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.108.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: \(x^{2} + 16 x + 3\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 8 a + 7 + \left(3 a + 11\right)\cdot 17 + \left(3 a + 4\right)\cdot 17^{2} + \left(16 a + 1\right)\cdot 17^{3} + \left(15 a + 10\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 5 + 17 + 9\cdot 17^{2} + 15\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 15 + \left(13 a + 6\right)\cdot 17 + \left(13 a + 4\right)\cdot 17^{2} + 14\cdot 17^{3} + \left(a + 9\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 9 a + 10 + \left(13 a + 5\right)\cdot 17 + \left(13 a + 12\right)\cdot 17^{2} + 15\cdot 17^{3} + \left(a + 6\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 12 + 15\cdot 17 + 7\cdot 17^{2} + 17^{3} + 14\cdot 17^{4} +O(17^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + 2 + \left(3 a + 10\right)\cdot 17 + \left(3 a + 12\right)\cdot 17^{2} + \left(16 a + 2\right)\cdot 17^{3} + \left(15 a + 7\right)\cdot 17^{4} +O(17^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.