# Properties

 Label 4.16400.6t13.d.a Dimension $4$ Group $C_3^2:D_4$ Conductor $16400$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$16400$$$$\medspace = 2^{4} \cdot 5^{2} \cdot 41$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.0.65600.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Determinant: 1.41.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.0.65600.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$19 a + 29 + \left(16 a + 5\right)\cdot 37 + \left(9 a + 36\right)\cdot 37^{2} + \left(7 a + 2\right)\cdot 37^{3} + \left(10 a + 36\right)\cdot 37^{4} +O(37^{5})$$ 19*a + 29 + (16*a + 5)*37 + (9*a + 36)*37^2 + (7*a + 2)*37^3 + (10*a + 36)*37^4+O(37^5) $r_{ 2 }$ $=$ $$23 a + 34 + \left(17 a + 24\right)\cdot 37 + \left(13 a + 35\right)\cdot 37^{2} + 17 a\cdot 37^{3} + \left(18 a + 7\right)\cdot 37^{4} +O(37^{5})$$ 23*a + 34 + (17*a + 24)*37 + (13*a + 35)*37^2 + 17*a*37^3 + (18*a + 7)*37^4+O(37^5) $r_{ 3 }$ $=$ $$18 a + 31 + \left(20 a + 15\right)\cdot 37 + \left(27 a + 20\right)\cdot 37^{2} + \left(29 a + 22\right)\cdot 37^{3} + \left(26 a + 32\right)\cdot 37^{4} +O(37^{5})$$ 18*a + 31 + (20*a + 15)*37 + (27*a + 20)*37^2 + (29*a + 22)*37^3 + (26*a + 32)*37^4+O(37^5) $r_{ 4 }$ $=$ $$26 + 13\cdot 37 + 3\cdot 37^{2} + 16\cdot 37^{3} + 3\cdot 37^{4} +O(37^{5})$$ 26 + 13*37 + 3*37^2 + 16*37^3 + 3*37^4+O(37^5) $r_{ 5 }$ $=$ $$15 + 15\cdot 37 + 17\cdot 37^{2} + 11\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})$$ 15 + 15*37 + 17*37^2 + 11*37^3 + 5*37^4+O(37^5) $r_{ 6 }$ $=$ $$14 a + 15 + \left(19 a + 35\right)\cdot 37 + \left(23 a + 34\right)\cdot 37^{2} + \left(19 a + 19\right)\cdot 37^{3} + \left(18 a + 26\right)\cdot 37^{4} +O(37^{5})$$ 14*a + 15 + (19*a + 35)*37 + (23*a + 34)*37^2 + (19*a + 19)*37^3 + (18*a + 26)*37^4+O(37^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.