# Properties

 Label 4.1516563.12t34.b Dimension $4$ Group $C_3^2:D_4$ Conductor $1516563$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$1516563$$$$\medspace = 3^{5} \cdot 79^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.57591.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: odd Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.0.57591.1

## Galois action

### Roots of defining polynomial

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:
 $r_{ 1 }$ $=$ $$27 + 20\cdot 31 + 29\cdot 31^{2} + 10\cdot 31^{3} + 16\cdot 31^{4} +O(31^{5})$$ 27 + 20*31 + 29*31^2 + 10*31^3 + 16*31^4+O(31^5) $r_{ 2 }$ $=$ $$24 a + 9 + \left(28 a + 19\right)\cdot 31 + \left(12 a + 17\right)\cdot 31^{2} + \left(6 a + 25\right)\cdot 31^{3} + \left(16 a + 9\right)\cdot 31^{4} +O(31^{5})$$ 24*a + 9 + (28*a + 19)*31 + (12*a + 17)*31^2 + (6*a + 25)*31^3 + (16*a + 9)*31^4+O(31^5) $r_{ 3 }$ $=$ $$16 + 31 + 8\cdot 31^{2} + 10\cdot 31^{3} + 19\cdot 31^{4} +O(31^{5})$$ 16 + 31 + 8*31^2 + 10*31^3 + 19*31^4+O(31^5) $r_{ 4 }$ $=$ $$6 a + 17 + \left(8 a + 9\right)\cdot 31 + 15 a\cdot 31^{2} + \left(28 a + 5\right)\cdot 31^{3} + \left(5 a + 14\right)\cdot 31^{4} +O(31^{5})$$ 6*a + 17 + (8*a + 9)*31 + 15*a*31^2 + (28*a + 5)*31^3 + (5*a + 14)*31^4+O(31^5) $r_{ 5 }$ $=$ $$7 a + 26 + \left(2 a + 21\right)\cdot 31 + \left(18 a + 14\right)\cdot 31^{2} + \left(24 a + 25\right)\cdot 31^{3} + \left(14 a + 4\right)\cdot 31^{4} +O(31^{5})$$ 7*a + 26 + (2*a + 21)*31 + (18*a + 14)*31^2 + (24*a + 25)*31^3 + (14*a + 4)*31^4+O(31^5) $r_{ 6 }$ $=$ $$25 a + 29 + \left(22 a + 19\right)\cdot 31 + \left(15 a + 22\right)\cdot 31^{2} + \left(2 a + 15\right)\cdot 31^{3} + \left(25 a + 28\right)\cdot 31^{4} +O(31^{5})$$ 25*a + 29 + (22*a + 19)*31 + (15*a + 22)*31^2 + (2*a + 15)*31^3 + (25*a + 28)*31^4+O(31^5)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $4$ $6$ $2$ $(1,3)(2,4)(5,6)$ $-2$ $6$ $2$ $(2,5)$ $0$ $9$ $2$ $(2,5)(4,6)$ $0$ $4$ $3$ $(1,2,5)$ $-2$ $4$ $3$ $(1,2,5)(3,4,6)$ $1$ $18$ $4$ $(1,3)(2,6,5,4)$ $0$ $12$ $6$ $(1,4,2,6,5,3)$ $1$ $12$ $6$ $(2,5)(3,4,6)$ $0$
The blue line marks the conjugacy class containing complex conjugation.