Properties

Label 2.2888.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $2888$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(2888\)\(\medspace = 2^{3} \cdot 19^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.158470336.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.8.2t1.b.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.2888.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 3x^{5} - x^{4} - 12x^{3} + 52x^{2} + 20x + 68 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: \( x^{2} + 6x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 3 a + \left(3 a + 2\right)\cdot 7 + \left(2 a + 1\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} + 6 a\cdot 7^{5} + \left(3 a + 2\right)\cdot 7^{6} + \left(a + 2\right)\cdot 7^{7} + 6 a\cdot 7^{8} + a\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 4 a + 3 + \left(3 a + 2\right)\cdot 7 + 4 a\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + 7^{5} + \left(3 a + 6\right)\cdot 7^{6} + \left(5 a + 6\right)\cdot 7^{7} + 4\cdot 7^{8} + \left(5 a + 2\right)\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 4 + 2\cdot 7^{2} + 3\cdot 7^{3} + 4\cdot 7^{4} + 6\cdot 7^{5} + 2\cdot 7^{6} + 2\cdot 7^{7} + 7^{8} + 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 5 + 2\cdot 7 + 4\cdot 7^{2} + 3\cdot 7^{3} + 5\cdot 7^{4} + 2\cdot 7^{5} + 2\cdot 7^{6} + 3\cdot 7^{8} + 5\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 5 a + 2\cdot 7 + \left(5 a + 4\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(6 a + 4\right)\cdot 7^{4} + \left(a + 3\right)\cdot 7^{5} + \left(2 a + 3\right)\cdot 7^{6} + 5\cdot 7^{7} + \left(4 a + 3\right)\cdot 7^{8} + \left(a + 3\right)\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 2 a + 5 + \left(6 a + 4\right)\cdot 7 + \left(a + 1\right)\cdot 7^{2} + 6\cdot 7^{3} + 4\cdot 7^{4} + \left(5 a + 5\right)\cdot 7^{5} + \left(4 a + 3\right)\cdot 7^{6} + \left(6 a + 3\right)\cdot 7^{7} + 2 a\cdot 7^{8} + \left(5 a + 1\right)\cdot 7^{9} +O(7^{10})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.