Properties

Label 2.87.6t3.b
Dimension $2$
Group $D_{6}$
Conductor $87$
Indicator $1$

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Basic invariants

Dimension:$2$
Group:$D_{6}$
Conductor:\(87\)\(\medspace = 3 \cdot 29 \)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin number field: Galois closure of 6.0.22707.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Projective image: $S_3$
Projective field: 3.1.87.1

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 $\mathbb{Q}_{ 19 }$: \(x^{2} + 18 x + 2\)  Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 6 a + 8 + \left(7 a + 12\right)\cdot 19 + \left(15 a + 9\right)\cdot 19^{2} + \left(3 a + 10\right)\cdot 19^{3} + \left(9 a + 18\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 11 a + 1 + \left(2 a + 8\right)\cdot 19 + 6 a\cdot 19^{2} + \left(16 a + 17\right)\cdot 19^{3} + \left(16 a + 13\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 18 + 14\cdot 19 + 6\cdot 19^{2} + 12\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 8\cdot 19 + 18\cdot 19^{2} + 9\cdot 19^{3} + 15\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 13 a + 14 + \left(11 a + 13\right)\cdot 19 + \left(3 a + 17\right)\cdot 19^{2} + \left(15 a + 17\right)\cdot 19^{3} + \left(9 a + 4\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 8 a + 12 + \left(16 a + 18\right)\cdot 19 + \left(12 a + 3\right)\cdot 19^{2} + \left(2 a + 8\right)\cdot 19^{3} + \left(2 a + 14\right)\cdot 19^{4} +O(19^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

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