Properties

Label 2.588.6t3.b.a
Dimension $2$
Group $D_{6}$
Conductor $588$
Root number $1$
Indicator $1$

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Basic invariants

Dimension: $2$
Group: $D_{6}$
Conductor: \(588\)\(\medspace = 2^{2} \cdot 3 \cdot 7^{2}\)
Frobenius-Schur indicator: $1$
Root number: $1$
Artin stem field: Galois closure of 6.0.2420208.1
Galois orbit size: $1$
Smallest permutation container: $D_{6}$
Parity: odd
Determinant: 1.3.2t1.a.a
Projective image: $S_3$
Projective stem field: Galois closure of 3.1.588.1

Defining polynomial

$f(x)$$=$ \( x^{6} - 2x^{5} - 3x^{4} + 6x^{3} + 9x^{2} - 4x + 1 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 7.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: \( x^{2} + 7x + 2 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 2 a + 7 a\cdot 11 + 8\cdot 11^{2} + \left(2 a + 10\right)\cdot 11^{3} + \left(7 a + 4\right)\cdot 11^{4} + \left(2 a + 6\right)\cdot 11^{5} + \left(8 a + 4\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 7 a + 4 + \left(6 a + 7\right)\cdot 11 + 4 a\cdot 11^{2} + \left(a + 1\right)\cdot 11^{3} + \left(8 a + 1\right)\cdot 11^{4} + \left(7 a + 9\right)\cdot 11^{5} + \left(a + 6\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 9 a + 8 + \left(3 a + 4\right)\cdot 11 + \left(10 a + 3\right)\cdot 11^{2} + \left(8 a + 7\right)\cdot 11^{3} + \left(3 a + 9\right)\cdot 11^{4} + \left(8 a + 9\right)\cdot 11^{5} + \left(2 a + 1\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 4 + 6\cdot 11 + 10\cdot 11^{2} + 3\cdot 11^{3} + 7\cdot 11^{4} + 5\cdot 11^{5} + 4\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 9 + 9\cdot 11 + 8\cdot 11^{2} + 7\cdot 11^{3} + 10\cdot 11^{4} + 2\cdot 11^{5} + 9\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 4 a + 10 + \left(4 a + 4\right)\cdot 11 + \left(6 a + 1\right)\cdot 11^{2} + \left(9 a + 2\right)\cdot 11^{3} + \left(2 a + 10\right)\cdot 11^{4} + \left(3 a + 9\right)\cdot 11^{5} + \left(9 a + 5\right)\cdot 11^{6} +O(11^{7})\) Copy content Toggle raw display

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.