# Properties

 Label 3.5808.6t11.a Dimension $3$ Group $S_4\times C_2$ Conductor $5808$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4\times C_2$ Conductor: $$5808$$$$\medspace = 2^{4} \cdot 3 \cdot 11^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.2811072.1 Galois orbit size: $1$ Smallest permutation container: $S_4\times C_2$ Parity: even Projective image: $S_4$ Projective field: Galois closure of 4.2.17424.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$
Roots:
 $r_{ 1 }$ $=$ $$2\cdot 7 + 6\cdot 7^{2} + 2\cdot 7^{3} + 5\cdot 7^{4} + 2\cdot 7^{5} + 4\cdot 7^{6} + 4\cdot 7^{7} + 6\cdot 7^{8} + 7^{9} +O(7^{10})$$ 2*7 + 6*7^2 + 2*7^3 + 5*7^4 + 2*7^5 + 4*7^6 + 4*7^7 + 6*7^8 + 7^9+O(7^10) $r_{ 2 }$ $=$ $$a + 4 + \left(6 a + 3\right)\cdot 7 + \left(2 a + 6\right)\cdot 7^{2} + \left(5 a + 2\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + 4 a\cdot 7^{5} + 6 a\cdot 7^{6} + \left(6 a + 3\right)\cdot 7^{7} + \left(5 a + 1\right)\cdot 7^{8} + \left(6 a + 1\right)\cdot 7^{9} +O(7^{10})$$ a + 4 + (6*a + 3)*7 + (2*a + 6)*7^2 + (5*a + 2)*7^3 + (a + 5)*7^4 + 4*a*7^5 + 6*a*7^6 + (6*a + 3)*7^7 + (5*a + 1)*7^8 + (6*a + 1)*7^9+O(7^10) $r_{ 3 }$ $=$ $$3 + 7 + 6\cdot 7^{2} + 2\cdot 7^{4} + 6\cdot 7^{5} + 3\cdot 7^{6} + 4\cdot 7^{7} + 3\cdot 7^{8} + 3\cdot 7^{9} +O(7^{10})$$ 3 + 7 + 6*7^2 + 2*7^4 + 6*7^5 + 3*7^6 + 4*7^7 + 3*7^8 + 3*7^9+O(7^10) $r_{ 4 }$ $=$ $$6 a + 5 + 7 + \left(4 a + 3\right)\cdot 7^{2} + \left(a + 5\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(2 a + 3\right)\cdot 7^{5} + 2\cdot 7^{6} + 3\cdot 7^{7} + a\cdot 7^{8} + 2\cdot 7^{9} +O(7^{10})$$ 6*a + 5 + 7 + (4*a + 3)*7^2 + (a + 5)*7^3 + (5*a + 1)*7^4 + (2*a + 3)*7^5 + 2*7^6 + 3*7^7 + a*7^8 + 2*7^9+O(7^10) $r_{ 5 }$ $=$ $$4 a + \left(4 a + 6\right)\cdot 7 + 4\cdot 7^{2} + \left(2 a + 3\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(4 a + 4\right)\cdot 7^{5} + \left(5 a + 4\right)\cdot 7^{6} + 2 a\cdot 7^{7} + \left(5 a + 3\right)\cdot 7^{8} + \left(6 a + 5\right)\cdot 7^{9} +O(7^{10})$$ 4*a + (4*a + 6)*7 + 4*7^2 + (2*a + 3)*7^3 + (5*a + 1)*7^4 + (4*a + 4)*7^5 + (5*a + 4)*7^6 + 2*a*7^7 + (5*a + 3)*7^8 + (6*a + 5)*7^9+O(7^10) $r_{ 6 }$ $=$ $$3 a + 4 + \left(2 a + 6\right)\cdot 7 + 6 a\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(a + 4\right)\cdot 7^{4} + \left(2 a + 3\right)\cdot 7^{5} + \left(a + 5\right)\cdot 7^{6} + \left(4 a + 4\right)\cdot 7^{7} + \left(a + 5\right)\cdot 7^{8} + 6\cdot 7^{9} +O(7^{10})$$ 3*a + 4 + (2*a + 6)*7 + 6*a*7^2 + (4*a + 5)*7^3 + (a + 4)*7^4 + (2*a + 3)*7^5 + (a + 5)*7^6 + (4*a + 4)*7^7 + (a + 5)*7^8 + 6*7^9+O(7^10)

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

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

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character values $c1$ $1$ $1$ $()$ $3$ $1$ $2$ $(1,3)(2,6)(4,5)$ $-3$ $3$ $2$ $(1,3)$ $1$ $3$ $2$ $(1,3)(2,6)$ $-1$ $6$ $2$ $(2,4)(5,6)$ $1$ $6$ $2$ $(1,3)(2,4)(5,6)$ $-1$ $8$ $3$ $(1,2,4)(3,6,5)$ $0$ $6$ $4$ $(1,6,3,2)$ $1$ $6$ $4$ $(1,3)(2,4,6,5)$ $-1$ $8$ $6$ $(1,6,5,3,2,4)$ $0$
The blue line marks the conjugacy class containing complex conjugation.