Properties

Label 2.1840.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $1840$
Indicator $1$

Related objects

Learn more

Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(1840\)\(\medspace = 2^{4} \cdot 5 \cdot 23 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.16928000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.460.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: \(x^{2} + 45 x + 5\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 5 + 18\cdot 47^{2} + 11\cdot 47^{3} + 6\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 27 + 38\cdot 47 + 37\cdot 47^{2} + 28\cdot 47^{3} + 16\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 43 a + 5 + \left(33 a + 4\right)\cdot 47 + \left(21 a + 28\right)\cdot 47^{2} + \left(5 a + 36\right)\cdot 47^{3} + \left(42 a + 31\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 44 a + 33 + \left(38 a + 17\right)\cdot 47 + \left(2 a + 26\right)\cdot 47^{2} + \left(15 a + 5\right)\cdot 47^{3} + \left(36 a + 6\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 3 a + 27 + \left(8 a + 4\right)\cdot 47 + \left(44 a + 40\right)\cdot 47^{2} + \left(31 a + 32\right)\cdot 47^{3} + \left(10 a + 16\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 44 + \left(13 a + 28\right)\cdot 47 + \left(25 a + 37\right)\cdot 47^{2} + \left(41 a + 25\right)\cdot 47^{3} + \left(4 a + 16\right)\cdot 47^{4} +O(47^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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