# Properties

 Label 4.25200125.12t34.a Dimension $4$ Group $C_3^2:D_4$ Conductor $25200125$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $4$ Group: $C_3^2:D_4$ Conductor: $$25200125$$$$\medspace = 5^{3} \cdot 449^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.2.56125.1 Galois orbit size: $1$ Smallest permutation container: 12T34 Parity: even Projective image: $\SOPlus(4,2)$ Projective field: Galois closure of 6.2.56125.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $$x^{2} + 38x + 6$$
Roots:
 $r_{ 1 }$ $=$ $$10 a + 19 + \left(19 a + 3\right)\cdot 41 + \left(28 a + 6\right)\cdot 41^{2} + \left(4 a + 35\right)\cdot 41^{3} + \left(36 a + 26\right)\cdot 41^{4} +O(41^{5})$$ 10*a + 19 + (19*a + 3)*41 + (28*a + 6)*41^2 + (4*a + 35)*41^3 + (36*a + 26)*41^4+O(41^5) $r_{ 2 }$ $=$ $$35 a + 32 + \left(25 a + 6\right)\cdot 41 + \left(30 a + 39\right)\cdot 41^{2} + \left(18 a + 20\right)\cdot 41^{3} + \left(3 a + 14\right)\cdot 41^{4} +O(41^{5})$$ 35*a + 32 + (25*a + 6)*41 + (30*a + 39)*41^2 + (18*a + 20)*41^3 + (3*a + 14)*41^4+O(41^5) $r_{ 3 }$ $=$ $$21 + 8\cdot 41 + 23\cdot 41^{2} + 36\cdot 41^{3} + 34\cdot 41^{4} +O(41^{5})$$ 21 + 8*41 + 23*41^2 + 36*41^3 + 34*41^4+O(41^5) $r_{ 4 }$ $=$ $$30 + 3\cdot 41 + 4\cdot 41^{3} + 33\cdot 41^{4} +O(41^{5})$$ 30 + 3*41 + 4*41^3 + 33*41^4+O(41^5) $r_{ 5 }$ $=$ $$31 a + 8 + \left(21 a + 10\right)\cdot 41 + \left(12 a + 31\right)\cdot 41^{2} + \left(36 a + 20\right)\cdot 41^{3} + \left(4 a + 7\right)\cdot 41^{4} +O(41^{5})$$ 31*a + 8 + (21*a + 10)*41 + (12*a + 31)*41^2 + (36*a + 20)*41^3 + (4*a + 7)*41^4+O(41^5) $r_{ 6 }$ $=$ $$6 a + 14 + \left(15 a + 8\right)\cdot 41 + \left(10 a + 23\right)\cdot 41^{2} + \left(22 a + 5\right)\cdot 41^{3} + \left(37 a + 6\right)\cdot 41^{4} +O(41^{5})$$ 6*a + 14 + (15*a + 8)*41 + (10*a + 23)*41^2 + (22*a + 5)*41^3 + (37*a + 6)*41^4+O(41^5)

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