Properties

Label 2.1080.6t3.f.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 6.2.5832000.1
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)$$=$ \( x^{6} - 3x^{4} - 9x^{2} - 30x - 20 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 16 + 14\cdot 19 + 5\cdot 19^{2} + 4\cdot 19^{3} + 13\cdot 19^{4} + 17\cdot 19^{5} +O(19^{6})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 6 a + 8 + \left(7 a + 1\right)\cdot 19 + \left(9 a + 15\right)\cdot 19^{2} + 3 a\cdot 19^{3} + \left(16 a + 6\right)\cdot 19^{4} + \left(5 a + 15\right)\cdot 19^{5} +O(19^{6})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 + 8\cdot 19 + 3\cdot 19^{2} + 10\cdot 19^{3} + 5\cdot 19^{4} + 5\cdot 19^{5} +O(19^{6})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 3 a + 13 + \left(8 a + 2\right)\cdot 19 + \left(10 a + 16\right)\cdot 19^{2} + \left(5 a + 6\right)\cdot 19^{3} + \left(17 a + 10\right)\cdot 19^{4} + \left(9 a + 10\right)\cdot 19^{5} +O(19^{6})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 16 a + 16 + \left(10 a + 7\right)\cdot 19 + \left(8 a + 18\right)\cdot 19^{2} + \left(13 a + 1\right)\cdot 19^{3} + \left(a + 3\right)\cdot 19^{4} + \left(9 a + 3\right)\cdot 19^{5} +O(19^{6})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 13 a + 14 + \left(11 a + 2\right)\cdot 19 + \left(9 a + 17\right)\cdot 19^{2} + \left(15 a + 13\right)\cdot 19^{3} + \left(2 a + 18\right)\cdot 19^{4} + \left(13 a + 4\right)\cdot 19^{5} +O(19^{6})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.