# Properties

 Label 4.16400.6t13.c.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 6.2.82000.1 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 6.2.82000.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{2} + 29x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$a + 25 + \left(23 a + 16\right)\cdot 31 + \left(10 a + 8\right)\cdot 31^{2} + \left(25 a + 15\right)\cdot 31^{3} + \left(13 a + 8\right)\cdot 31^{4} +O(31^{5})$$ a + 25 + (23*a + 16)*31 + (10*a + 8)*31^2 + (25*a + 15)*31^3 + (13*a + 8)*31^4+O(31^5) $r_{ 2 }$ $=$ $$30 a + 27 + \left(7 a + 30\right)\cdot 31 + \left(20 a + 6\right)\cdot 31^{2} + \left(5 a + 24\right)\cdot 31^{3} + \left(17 a + 10\right)\cdot 31^{4} +O(31^{5})$$ 30*a + 27 + (7*a + 30)*31 + (20*a + 6)*31^2 + (5*a + 24)*31^3 + (17*a + 10)*31^4+O(31^5) $r_{ 3 }$ $=$ $$10 a + 4 + \left(4 a + 6\right)\cdot 31 + \left(23 a + 29\right)\cdot 31^{2} + \left(8 a + 5\right)\cdot 31^{3} + \left(21 a + 23\right)\cdot 31^{4} +O(31^{5})$$ 10*a + 4 + (4*a + 6)*31 + (23*a + 29)*31^2 + (8*a + 5)*31^3 + (21*a + 23)*31^4+O(31^5) $r_{ 4 }$ $=$ $$21 a + 24 + \left(26 a + 4\right)\cdot 31 + \left(7 a + 9\right)\cdot 31^{2} + 22 a\cdot 31^{3} + \left(9 a + 26\right)\cdot 31^{4} +O(31^{5})$$ 21*a + 24 + (26*a + 4)*31 + (7*a + 9)*31^2 + 22*a*31^3 + (9*a + 26)*31^4+O(31^5) $r_{ 5 }$ $=$ $$22 + 13\cdot 31 + 11\cdot 31^{2} + 24\cdot 31^{3} + 13\cdot 31^{4} +O(31^{5})$$ 22 + 13*31 + 11*31^2 + 24*31^3 + 13*31^4+O(31^5) $r_{ 6 }$ $=$ $$23 + 20\cdot 31 + 27\cdot 31^{2} + 22\cdot 31^{3} + 10\cdot 31^{4} +O(31^{5})$$ 23 + 20*31 + 27*31^2 + 22*31^3 + 10*31^4+O(31^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.