Properties

 Label 4.449307.6t13.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $449307$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$449307$$$$\medspace = 3^{5} \cdot 43^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.31347.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.0.31347.1

Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$12 a + 2 + \left(7 a + 11\right)\cdot 13 + \left(5 a + 4\right)\cdot 13^{2} + \left(5 a + 1\right)\cdot 13^{3} + \left(4 a + 9\right)\cdot 13^{4} +O(13^{5})$$ 12*a + 2 + (7*a + 11)*13 + (5*a + 4)*13^2 + (5*a + 1)*13^3 + (4*a + 9)*13^4+O(13^5) $r_{ 2 }$ $=$ $$a + 7 + \left(a + 1\right)\cdot 13 + \left(7 a + 10\right)\cdot 13^{2} + \left(3 a + 11\right)\cdot 13^{3} + \left(6 a + 8\right)\cdot 13^{4} +O(13^{5})$$ a + 7 + (a + 1)*13 + (7*a + 10)*13^2 + (3*a + 11)*13^3 + (6*a + 8)*13^4+O(13^5) $r_{ 3 }$ $=$ $$11 + 9\cdot 13 + 12\cdot 13^{2} + 5\cdot 13^{3} + 5\cdot 13^{4} +O(13^{5})$$ 11 + 9*13 + 12*13^2 + 5*13^3 + 5*13^4+O(13^5) $r_{ 4 }$ $=$ $$a + 1 + \left(5 a + 7\right)\cdot 13 + \left(7 a + 2\right)\cdot 13^{2} + \left(7 a + 1\right)\cdot 13^{3} + \left(8 a + 8\right)\cdot 13^{4} +O(13^{5})$$ a + 1 + (5*a + 7)*13 + (7*a + 2)*13^2 + (7*a + 1)*13^3 + (8*a + 8)*13^4+O(13^5) $r_{ 5 }$ $=$ $$10 + 7\cdot 13 + 5\cdot 13^{2} + 10\cdot 13^{3} + 8\cdot 13^{4} +O(13^{5})$$ 10 + 7*13 + 5*13^2 + 10*13^3 + 8*13^4+O(13^5) $r_{ 6 }$ $=$ $$12 a + 8 + \left(11 a + 1\right)\cdot 13 + \left(5 a + 3\right)\cdot 13^{2} + \left(9 a + 8\right)\cdot 13^{3} + \left(6 a + 11\right)\cdot 13^{4} +O(13^{5})$$ 12*a + 8 + (11*a + 1)*13 + (5*a + 3)*13^2 + (9*a + 8)*13^3 + (6*a + 11)*13^4+O(13^5)

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $2$ $6$ $2$ $(3,6)$ $0$ $9$ $2$ $(3,6)(4,5)$ $0$ $4$ $3$ $(1,4,5)(2,3,6)$ $1$ $4$ $3$ $(1,4,5)$ $-2$ $18$ $4$ $(1,2)(3,5,6,4)$ $0$ $12$ $6$ $(1,3,4,6,5,2)$ $-1$ $12$ $6$ $(1,4,5)(3,6)$ $0$

The blue line marks the conjugacy class containing complex conjugation.