# Properties

 Label 4.580765248.12t34.i.a Dimension $4$ Group $C_3^2:D_4$ Conductor $580765248$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$580765248$$$$\medspace = 2^{6} \cdot 3^{3} \cdot 7^{2} \cdot 19^{3}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 6.2.20741616.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: even Determinant: 1.57.2t1.a.a Projective image: $\SOPlus(4,2)$ Projective stem field: Galois closure of 6.2.20741616.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 6x^{4} + 14x^{2} + 16x + 4$$ x^6 - x^5 - 6*x^4 + 14*x^2 + 16*x + 4 .

The roots of $f$ are computed in an extension of $\Q_{ 113 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 113 }$: $$x^{2} + 101x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$88 a + 34 + \left(76 a + 104\right)\cdot 113 + \left(76 a + 100\right)\cdot 113^{2} + \left(64 a + 102\right)\cdot 113^{3} + \left(54 a + 28\right)\cdot 113^{4} +O(113^{5})$$ 88*a + 34 + (76*a + 104)*113 + (76*a + 100)*113^2 + (64*a + 102)*113^3 + (54*a + 28)*113^4+O(113^5) $r_{ 2 }$ $=$ $$36 a + 106 + \left(100 a + 23\right)\cdot 113 + \left(17 a + 105\right)\cdot 113^{2} + \left(45 a + 5\right)\cdot 113^{3} + \left(111 a + 57\right)\cdot 113^{4} +O(113^{5})$$ 36*a + 106 + (100*a + 23)*113 + (17*a + 105)*113^2 + (45*a + 5)*113^3 + (111*a + 57)*113^4+O(113^5) $r_{ 3 }$ $=$ $$104 + 98\cdot 113 + 55\cdot 113^{2} + 98\cdot 113^{3} + 76\cdot 113^{4} +O(113^{5})$$ 104 + 98*113 + 55*113^2 + 98*113^3 + 76*113^4+O(113^5) $r_{ 4 }$ $=$ $$50 + 16\cdot 113 + 43\cdot 113^{2} + 42\cdot 113^{3} + 16\cdot 113^{4} +O(113^{5})$$ 50 + 16*113 + 43*113^2 + 42*113^3 + 16*113^4+O(113^5) $r_{ 5 }$ $=$ $$77 a + 86 + \left(12 a + 61\right)\cdot 113 + \left(95 a + 106\right)\cdot 113^{2} + \left(67 a + 77\right)\cdot 113^{3} + \left(a + 105\right)\cdot 113^{4} +O(113^{5})$$ 77*a + 86 + (12*a + 61)*113 + (95*a + 106)*113^2 + (67*a + 77)*113^3 + (a + 105)*113^4+O(113^5) $r_{ 6 }$ $=$ $$25 a + 73 + \left(36 a + 33\right)\cdot 113 + \left(36 a + 40\right)\cdot 113^{2} + \left(48 a + 11\right)\cdot 113^{3} + \left(58 a + 54\right)\cdot 113^{4} +O(113^{5})$$ 25*a + 73 + (36*a + 33)*113 + (36*a + 40)*113^2 + (48*a + 11)*113^3 + (58*a + 54)*113^4+O(113^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.