# Properties

 Label 4.13401.6t13.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $13401$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$13401$$$$\medspace = 3^{2} \cdot 1489$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.40203.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Determinant: 1.1489.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.0.40203.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$8 a + 4 + \left(3 a + 8\right)\cdot 19 + 17\cdot 19^{2} + \left(13 a + 13\right)\cdot 19^{3} + \left(11 a + 14\right)\cdot 19^{4} +O(19^{5})$$ 8*a + 4 + (3*a + 8)*19 + 17*19^2 + (13*a + 13)*19^3 + (11*a + 14)*19^4+O(19^5) $r_{ 2 }$ $=$ $$15 + 10\cdot 19 + 17\cdot 19^{2} + 15\cdot 19^{3} + 8\cdot 19^{4} +O(19^{5})$$ 15 + 10*19 + 17*19^2 + 15*19^3 + 8*19^4+O(19^5) $r_{ 3 }$ $=$ $$11 + 13\cdot 19 + 14\cdot 19^{2} + 9\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})$$ 11 + 13*19 + 14*19^2 + 9*19^3 + 9*19^4+O(19^5) $r_{ 4 }$ $=$ $$11 a + 12 + \left(15 a + 3\right)\cdot 19 + \left(18 a + 14\right)\cdot 19^{2} + \left(5 a + 7\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} +O(19^{5})$$ 11*a + 12 + (15*a + 3)*19 + (18*a + 14)*19^2 + (5*a + 7)*19^3 + (7*a + 13)*19^4+O(19^5) $r_{ 5 }$ $=$ $$7 a + 14 + \left(9 a + 18\right)\cdot 19 + \left(7 a + 6\right)\cdot 19^{2} + 16 a\cdot 19^{3} + \left(2 a + 12\right)\cdot 19^{4} +O(19^{5})$$ 7*a + 14 + (9*a + 18)*19 + (7*a + 6)*19^2 + 16*a*19^3 + (2*a + 12)*19^4+O(19^5) $r_{ 6 }$ $=$ $$12 a + 2 + \left(9 a + 2\right)\cdot 19 + \left(11 a + 5\right)\cdot 19^{2} + \left(2 a + 9\right)\cdot 19^{3} + \left(16 a + 17\right)\cdot 19^{4} +O(19^{5})$$ 12*a + 2 + (9*a + 2)*19 + (11*a + 5)*19^2 + (2*a + 9)*19^3 + (16*a + 17)*19^4+O(19^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.