# Properties

 Label 2.44.3t2.a.a Dimension $2$ Group $S_3$ Conductor $44$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$44$$$$\medspace = 2^{2} \cdot 11$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin field: 6.0.21296.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.11.2t1.a.a Projective image: $S_3$ Projective field: 6.0.21296.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 2 x^{4} - 3 x^{3} + 2 x^{2} - x + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6 x + 3$$

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.