# Properties

 Label 2.243.3t2.b.a Dimension $2$ Group $S_3$ Conductor $243$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$243$$$$\medspace = 3^{5}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 3.1.243.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.243.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$4 + 47\cdot 61 + 43\cdot 61^{2} + 4\cdot 61^{3} + 39\cdot 61^{4} +O(61^{5})$$ 4 + 47*61 + 43*61^2 + 4*61^3 + 39*61^4+O(61^5) $r_{ 2 }$ $=$ $$5 + 26\cdot 61 + 49\cdot 61^{2} + 20\cdot 61^{3} + 32\cdot 61^{4} +O(61^{5})$$ 5 + 26*61 + 49*61^2 + 20*61^3 + 32*61^4+O(61^5) $r_{ 3 }$ $=$ $$52 + 48\cdot 61 + 28\cdot 61^{2} + 35\cdot 61^{3} + 50\cdot 61^{4} +O(61^{5})$$ 52 + 48*61 + 28*61^2 + 35*61^3 + 50*61^4+O(61^5)

## 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 value $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.