# Properties

 Label 4.7198203.6t13.b.a Dimension $4$ Group $C_3^2:D_4$ Conductor $7198203$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$7198203$$$$\medspace = 3 \cdot 1549^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.41823.1 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 6.0.41823.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.