Properties

Label 2.87.6t3.a.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: 6.2.219501.1
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)$$=$\(x^{6} - x^{5} - 3 x^{4} + 3 x^{2} + 4 x + 1\)  Toggle raw display.

The roots of $f$ are computed in an extension of $\Q_{ 23 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 23 }$: \(x^{2} + 21 x + 5\)  Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 21 a + 3 + \left(22 a + 9\right)\cdot 23 + \left(a + 20\right)\cdot 23^{2} + \left(3 a + 20\right)\cdot 23^{3} + \left(17 a + 21\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 2 }$ $=$ \( 4 + 5\cdot 23 + 11\cdot 23^{2} + 21\cdot 23^{3} + 18\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 3 }$ $=$ \( 2 a + 22 + 10\cdot 23 + \left(21 a + 1\right)\cdot 23^{2} + \left(19 a + 2\right)\cdot 23^{3} + \left(5 a + 7\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 4 }$ $=$ \( 19 a + 17 + \left(6 a + 22\right)\cdot 23 + \left(2 a + 5\right)\cdot 23^{2} + \left(2 a + 1\right)\cdot 23^{3} + \left(11 a + 21\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 5 }$ $=$ \( 4 a + 9 + \left(16 a + 17\right)\cdot 23 + \left(20 a + 3\right)\cdot 23^{2} + \left(20 a + 3\right)\cdot 23^{3} + \left(11 a + 18\right)\cdot 23^{4} +O(23^{5})\)  Toggle raw display
$r_{ 6 }$ $=$ \( 15 + 3\cdot 23 + 3\cdot 23^{2} + 20\cdot 23^{3} + 4\cdot 23^{4} +O(23^{5})\)  Toggle raw display

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

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

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 6 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,4)(2,6)(3,5)$$-2$
$3$$2$$(1,2)(4,6)$$0$
$3$$2$$(1,6)(2,4)(3,5)$$0$
$2$$3$$(1,3,2)(4,5,6)$$-1$
$2$$6$$(1,5,2,4,3,6)$$1$

The blue line marks the conjugacy class containing complex conjugation.