# Properties

 Label 4.363888.6t13.b Dimension $4$ Group $C_3^2:D_4$ Conductor $363888$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$363888$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 7 \cdot 19^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.20741616.1 Galois orbit size: $1$ Smallest permutation container: $C_3^2:D_4$ Parity: even Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.2.20741616.1

## Galois action

### Roots of defining polynomial

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 values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,2)(3,4)(5,6)$ $0$ $6$ $2$ $(2,4)$ $2$ $9$ $2$ $(1,3)(2,4)$ $0$ $4$ $3$ $(1,3,6)(2,4,5)$ $-2$ $4$ $3$ $(1,3,6)$ $1$ $18$ $4$ $(1,2,3,4)(5,6)$ $0$ $12$ $6$ $(1,4,3,5,6,2)$ $0$ $12$ $6$ $(1,3,6)(2,4)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.