# Properties

 Label 2.3639.6t3.a.a Dimension $2$ Group $D_{6}$ Conductor $3639$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{6}$ Conductor: $$3639$$$$\medspace = 3 \cdot 1213$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 6.2.16062935373.1 Galois orbit size: $1$ Smallest permutation container: $D_{6}$ Parity: odd Determinant: 1.3639.2t1.a.a Projective image: $S_3$ Projective stem field: 3.1.3639.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 44 x^{3} - 729$$  .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.