Properties

Label 2.140.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $140$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.2.98000.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: Galois closure of 3.1.140.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: \( x^{2} + 18x + 2 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 15 a + 1 + \left(12 a + 13\right)\cdot 19 + \left(4 a + 6\right)\cdot 19^{2} + \left(6 a + 15\right)\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 11 a + 2 + \left(16 a + 14\right)\cdot 19 + 5\cdot 19^{2} + \left(15 a + 15\right)\cdot 19^{3} + \left(18 a + 10\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 + 6\cdot 19 + 14\cdot 19^{2} + 7\cdot 19^{3} + 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 17 + 11\cdot 19 + 3\cdot 19^{2} + 10\cdot 19^{3} + 16\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 16 + \left(6 a + 10\right)\cdot 19 + \left(14 a + 17\right)\cdot 19^{2} + \left(12 a + 16\right)\cdot 19^{3} + \left(18 a + 3\right)\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + 13 + 2 a\cdot 19 + \left(18 a + 9\right)\cdot 19^{2} + \left(3 a + 10\right)\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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