# Properties

 Label 2.2888.6t3.b Dimension $2$ Group $D_{6}$ Conductor $2888$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$2888$$$$\medspace = 2^{3} \cdot 19^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.158470336.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.2888.1

## 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:
 $r_{ 1 }$ $=$ $$3 a + \left(3 a + 2\right)\cdot 7 + \left(2 a + 1\right)\cdot 7^{2} + \left(2 a + 5\right)\cdot 7^{3} + \left(5 a + 2\right)\cdot 7^{4} + 6 a\cdot 7^{5} + \left(3 a + 2\right)\cdot 7^{6} + \left(a + 2\right)\cdot 7^{7} + 6 a\cdot 7^{8} + a\cdot 7^{9} +O(7^{10})$$ 3*a + (3*a + 2)*7 + (2*a + 1)*7^2 + (2*a + 5)*7^3 + (5*a + 2)*7^4 + 6*a*7^5 + (3*a + 2)*7^6 + (a + 2)*7^7 + 6*a*7^8 + a*7^9+O(7^10) $r_{ 2 }$ $=$ $$4 a + 3 + \left(3 a + 2\right)\cdot 7 + 4 a\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + \left(a + 5\right)\cdot 7^{4} + 7^{5} + \left(3 a + 6\right)\cdot 7^{6} + \left(5 a + 6\right)\cdot 7^{7} + 4\cdot 7^{8} + \left(5 a + 2\right)\cdot 7^{9} +O(7^{10})$$ 4*a + 3 + (3*a + 2)*7 + 4*a*7^2 + (4*a + 5)*7^3 + (a + 5)*7^4 + 7^5 + (3*a + 6)*7^6 + (5*a + 6)*7^7 + 4*7^8 + (5*a + 2)*7^9+O(7^10) $r_{ 3 }$ $=$ $$4 + 2\cdot 7^{2} + 3\cdot 7^{3} + 4\cdot 7^{4} + 6\cdot 7^{5} + 2\cdot 7^{6} + 2\cdot 7^{7} + 7^{8} + 7^{9} +O(7^{10})$$ 4 + 2*7^2 + 3*7^3 + 4*7^4 + 6*7^5 + 2*7^6 + 2*7^7 + 7^8 + 7^9+O(7^10) $r_{ 4 }$ $=$ $$5 + 2\cdot 7 + 4\cdot 7^{2} + 3\cdot 7^{3} + 5\cdot 7^{4} + 2\cdot 7^{5} + 2\cdot 7^{6} + 3\cdot 7^{8} + 5\cdot 7^{9} +O(7^{10})$$ 5 + 2*7 + 4*7^2 + 3*7^3 + 5*7^4 + 2*7^5 + 2*7^6 + 3*7^8 + 5*7^9+O(7^10) $r_{ 5 }$ $=$ $$5 a + 2\cdot 7 + \left(5 a + 4\right)\cdot 7^{2} + \left(6 a + 4\right)\cdot 7^{3} + \left(6 a + 4\right)\cdot 7^{4} + \left(a + 3\right)\cdot 7^{5} + \left(2 a + 3\right)\cdot 7^{6} + 5\cdot 7^{7} + \left(4 a + 3\right)\cdot 7^{8} + \left(a + 3\right)\cdot 7^{9} +O(7^{10})$$ 5*a + 2*7 + (5*a + 4)*7^2 + (6*a + 4)*7^3 + (6*a + 4)*7^4 + (a + 3)*7^5 + (2*a + 3)*7^6 + 5*7^7 + (4*a + 3)*7^8 + (a + 3)*7^9+O(7^10) $r_{ 6 }$ $=$ $$2 a + 5 + \left(6 a + 4\right)\cdot 7 + \left(a + 1\right)\cdot 7^{2} + 6\cdot 7^{3} + 4\cdot 7^{4} + \left(5 a + 5\right)\cdot 7^{5} + \left(4 a + 3\right)\cdot 7^{6} + \left(6 a + 3\right)\cdot 7^{7} + 2 a\cdot 7^{8} + \left(5 a + 1\right)\cdot 7^{9} +O(7^{10})$$ 2*a + 5 + (6*a + 4)*7 + (a + 1)*7^2 + 6*7^3 + 4*7^4 + (5*a + 5)*7^5 + (4*a + 3)*7^6 + (6*a + 3)*7^7 + 2*a*7^8 + (5*a + 1)*7^9+O(7^10)

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

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

### Character values on conjugacy classes

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