Properties

Label 2.87.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $87$
Root number $1$
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 stem field: 6.0.22707.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.87.2t1.a.a
Projective image: $S_3$
Projective stem field: 3.1.87.1

Defining polynomial

$f(x)$$=$\(x^{6} - x^{5} + 4 x^{4} - 4 x^{3} + 5 x^{2} - 3 x + 1\)  Toggle raw display.

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 value
$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.