Properties

Label 2.1080.6t3.f
Dimension $2$
Group $D_{6}$
Conductor $1080$
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 number field: Galois closure of 6.2.5832000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.1080.1

Galois action

Roots of defining polynomial

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 values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,3)(2,5)(4,6)$ $-2$
$3$ $2$ $(1,2)(3,5)$ $0$
$3$ $2$ $(1,5)(2,3)(4,6)$ $0$
$2$ $3$ $(1,6,2)(3,4,5)$ $-1$
$2$ $6$ $(1,4,2,3,6,5)$ $1$
The blue line marks the conjugacy class containing complex conjugation.