Properties

Label 2.28096.6t3.a.a
Dimension $2$
Group $D_{6}$
Conductor $28096$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(28096\)\(\medspace = 2^{6} \cdot 439 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: 6.2.98673152.2
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.439.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.439.1

Defining polynomial

$f(x)$$=$\(x^{6} - 2 x^{5} - 9 x^{4} + 22 x^{3} + 2 x^{2} + 24 x + 9\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 7.

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

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.