Properties

Label 2.2160.6t3.d
Dimension $2$
Group $D_{6}$
Conductor $2160$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(2160\)\(\medspace = 2^{4} \cdot 3^{3} \cdot 5 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.3499200.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.135.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 4 + 8\cdot 11 + 8\cdot 11^{2} + 5\cdot 11^{3} + 7\cdot 11^{4} + 9\cdot 11^{5} + 7\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 8 a + \left(4 a + 3\right)\cdot 11 + 4 a\cdot 11^{2} + \left(9 a + 4\right)\cdot 11^{3} + \left(8 a + 6\right)\cdot 11^{4} + 4 a\cdot 11^{5} + \left(4 a + 9\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a + 10 + \left(6 a + 2\right)\cdot 11 + \left(6 a + 2\right)\cdot 11^{2} + \left(a + 4\right)\cdot 11^{3} + \left(2 a + 10\right)\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} + \left(6 a + 10\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 7 + 2\cdot 11 + 2\cdot 11^{2} + 5\cdot 11^{3} + 3\cdot 11^{4} + 11^{5} + 3\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + \left(6 a + 8\right)\cdot 11 + \left(6 a + 10\right)\cdot 11^{2} + \left(a + 6\right)\cdot 11^{3} + \left(2 a + 4\right)\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} + \left(6 a + 1\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + 1 + \left(4 a + 8\right)\cdot 11 + \left(4 a + 8\right)\cdot 11^{2} + \left(9 a + 6\right)\cdot 11^{3} + 8 a\cdot 11^{4} + 4 a\cdot 11^{5} + 4 a\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character values
$c1$
$1$ $1$ $()$ $2$
$1$ $2$ $(1,4)(2,5)(3,6)$ $-2$
$3$ $2$ $(2,3)(5,6)$ $0$
$3$ $2$ $(1,2)(3,6)(4,5)$ $0$
$2$ $3$ $(1,6,5)(2,4,3)$ $-1$
$2$ $6$ $(1,2,6,4,5,3)$ $1$
The blue line marks the conjugacy class containing complex conjugation.