# Properties

 Label 2.135.6t3.a.a Dimension $2$ Group $D_{6}$ Conductor $135$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$135$$$$\medspace = 3^{3} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 6.2.91125.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Determinant: 1.15.2t1.a.a Projective image: $S_3$ Projective stem field: 3.1.135.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{3} - 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$5 a + 4 + 10\cdot 11 + \left(5 a + 10\right)\cdot 11^{2} + \left(5 a + 2\right)\cdot 11^{3} + \left(8 a + 2\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 2 }$ $=$ $$2 + 7\cdot 11 + 6\cdot 11^{2} + 7\cdot 11^{3} + 3\cdot 11^{4} +O(11^{5})$$ $r_{ 3 }$ $=$ $$5 + 4\cdot 11 + 2\cdot 11^{2} + 10\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})$$ $r_{ 4 }$ $=$ $$9 a + 3 + \left(9 a + 3\right)\cdot 11 + \left(a + 3\right)\cdot 11^{2} + \left(8 a + 8\right)\cdot 11^{3} + \left(4 a + 3\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 5 }$ $=$ $$6 a + 2 + \left(10 a + 7\right)\cdot 11 + \left(5 a + 8\right)\cdot 11^{2} + \left(5 a + 8\right)\cdot 11^{3} + \left(2 a + 8\right)\cdot 11^{4} +O(11^{5})$$ $r_{ 6 }$ $=$ $$2 a + 6 + a\cdot 11 + \left(9 a + 1\right)\cdot 11^{2} + \left(2 a + 6\right)\cdot 11^{3} + \left(6 a + 3\right)\cdot 11^{4} +O(11^{5})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)(5,6)$ $-2$ $3$ $2$ $(1,2)(3,4)(5,6)$ $0$ $3$ $2$ $(1,5)(4,6)$ $0$ $2$ $3$ $(1,3,5)(2,6,4)$ $-1$ $2$ $6$ $(1,6,3,4,5,2)$ $1$

The blue line marks the conjugacy class containing complex conjugation.