# Properties

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

# Related objects

## 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

 $f(x)$ $=$ $$x^{6} - 2x^{5} - 3x^{4} + 6x^{3} + 9x^{2} - 4x + 1$$ x^6 - 2*x^5 - 3*x^4 + 6*x^3 + 9*x^2 - 4*x + 1 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

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

## 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

 Size Order Action 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.