# Properties

 Label 2.1840.6t3.a Dimension $2$ Group $D_{6}$ Conductor $1840$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$1840$$$$\medspace = 2^{4} \cdot 5 \cdot 23$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 6.0.3385600.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.460.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12x + 2$$
Roots:
 $r_{ 1 }$ $=$ $$9 a + 11 + \left(a + 9\right)\cdot 13 + \left(3 a + 3\right)\cdot 13^{2} + 5 a\cdot 13^{3} + \left(a + 1\right)\cdot 13^{4} + \left(7 a + 1\right)\cdot 13^{5} + \left(7 a + 11\right)\cdot 13^{6} +O(13^{7})$$ 9*a + 11 + (a + 9)*13 + (3*a + 3)*13^2 + 5*a*13^3 + (a + 1)*13^4 + (7*a + 1)*13^5 + (7*a + 11)*13^6+O(13^7) $r_{ 2 }$ $=$ $$8 + 4\cdot 13^{2} + 10\cdot 13^{3} + 13^{4} + 5\cdot 13^{5} + 3\cdot 13^{6} +O(13^{7})$$ 8 + 4*13^2 + 10*13^3 + 13^4 + 5*13^5 + 3*13^6+O(13^7) $r_{ 3 }$ $=$ $$7 a + 10 + \left(11 a + 12\right)\cdot 13 + \left(5 a + 11\right)\cdot 13^{2} + \left(8 a + 8\right)\cdot 13^{3} + \left(9 a + 12\right)\cdot 13^{4} + \left(2 a + 1\right)\cdot 13^{5} + \left(9 a + 2\right)\cdot 13^{6} +O(13^{7})$$ 7*a + 10 + (11*a + 12)*13 + (5*a + 11)*13^2 + (8*a + 8)*13^3 + (9*a + 12)*13^4 + (2*a + 1)*13^5 + (9*a + 2)*13^6+O(13^7) $r_{ 4 }$ $=$ $$12 + 8\cdot 13 + 7\cdot 13^{2} + 5\cdot 13^{3} + 12\cdot 13^{4} + 2\cdot 13^{5} + 2\cdot 13^{6} +O(13^{7})$$ 12 + 8*13 + 7*13^2 + 5*13^3 + 12*13^4 + 2*13^5 + 2*13^6+O(13^7) $r_{ 5 }$ $=$ $$4 a + 7 + \left(11 a + 2\right)\cdot 13 + \left(9 a + 5\right)\cdot 13^{2} + \left(7 a + 2\right)\cdot 13^{3} + \left(11 a + 10\right)\cdot 13^{4} + \left(5 a + 6\right)\cdot 13^{5} + \left(5 a + 11\right)\cdot 13^{6} +O(13^{7})$$ 4*a + 7 + (11*a + 2)*13 + (9*a + 5)*13^2 + (7*a + 2)*13^3 + (11*a + 10)*13^4 + (5*a + 6)*13^5 + (5*a + 11)*13^6+O(13^7) $r_{ 6 }$ $=$ $$6 a + 4 + \left(a + 4\right)\cdot 13 + \left(7 a + 6\right)\cdot 13^{2} + \left(4 a + 11\right)\cdot 13^{3} + 3 a\cdot 13^{4} + \left(10 a + 8\right)\cdot 13^{5} + \left(3 a + 8\right)\cdot 13^{6} +O(13^{7})$$ 6*a + 4 + (a + 4)*13 + (7*a + 6)*13^2 + (4*a + 11)*13^3 + 3*a*13^4 + (10*a + 8)*13^5 + (3*a + 8)*13^6+O(13^7)

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

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

### Character values on conjugacy classes

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