# Properties

 Label 2.135.3t2.b Dimension $2$ Group $S_3$ Conductor $135$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$135$$$$\medspace = 3^{3} \cdot 5$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.135.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.135.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$11 + 17\cdot 47 + 27\cdot 47^{2} + 42\cdot 47^{3} + 3\cdot 47^{4} +O(47^{5})$$ 11 + 17*47 + 27*47^2 + 42*47^3 + 3*47^4+O(47^5) $r_{ 2 }$ $=$ $$41 + 38\cdot 47 + 43\cdot 47^{2} + 21\cdot 47^{3} + 15\cdot 47^{4} +O(47^{5})$$ 41 + 38*47 + 43*47^2 + 21*47^3 + 15*47^4+O(47^5) $r_{ 3 }$ $=$ $$42 + 37\cdot 47 + 22\cdot 47^{2} + 29\cdot 47^{3} + 27\cdot 47^{4} +O(47^{5})$$ 42 + 37*47 + 22*47^2 + 29*47^3 + 27*47^4+O(47^5)

### Generators of the action on the roots $r_{ 1 }, r_{ 2 }, r_{ 3 }$

 Cycle notation $(1,2,3)$ $(1,2)$

### Character values on conjugacy classes

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