Properties

Label 2.1728.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $1728$
Indicator $1$

Related objects

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(1728\)\(\medspace = 2^{6} \cdot 3^{3}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.1492992.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.108.1

Galois action

Roots of defining polynomial

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

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

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

Character values on conjugacy classes

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