Properties

Label 2.1080.6t3.d.a
Dimension $2$
Group $D_{6}$
Conductor $1080$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(1080\)\(\medspace = 2^{3} \cdot 3^{3} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.2.9331200.2
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.120.2t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.1080.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: \( x^{2} + 38x + 6 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 6 a + 17 + \left(21 a + 21\right)\cdot 41 + \left(11 a + 30\right)\cdot 41^{2} + \left(18 a + 18\right)\cdot 41^{3} + \left(28 a + 16\right)\cdot 41^{4} + \left(36 a + 14\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 a + 11 + \left(10 a + 2\right)\cdot 41 + \left(23 a + 13\right)\cdot 41^{2} + \left(29 a + 37\right)\cdot 41^{3} + \left(36 a + 9\right)\cdot 41^{4} + \left(35 a + 12\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 30 + 22\cdot 41 + 7\cdot 41^{2} + 41^{3} + 23\cdot 41^{4} + 12\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 35 a + 35 + \left(19 a + 37\right)\cdot 41 + \left(29 a + 2\right)\cdot 41^{2} + \left(22 a + 21\right)\cdot 41^{3} + \left(12 a + 1\right)\cdot 41^{4} + \left(4 a + 14\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 37 a + 23 + \left(30 a + 28\right)\cdot 41 + \left(17 a + 31\right)\cdot 41^{2} + \left(11 a + 20\right)\cdot 41^{3} + \left(4 a + 8\right)\cdot 41^{4} + \left(5 a + 1\right)\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 7 + 10\cdot 41 + 37\cdot 41^{2} + 23\cdot 41^{3} + 22\cdot 41^{4} + 27\cdot 41^{5} +O(41^{6})\) Copy content Toggle raw display

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

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

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$$2$$(2,6)(3,4)$$0$
$3$$2$$(1,2)(3,6)(4,5)$$0$
$2$$3$$(1,3,4)(2,5,6)$$-1$
$2$$6$$(1,2,3,5,4,6)$$1$

The blue line marks the conjugacy class containing complex conjugation.