# Properties

 Label 2.3332.3t2.a.a Dimension $2$ Group $S_3$ Conductor $3332$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$3332$$$$\medspace = 2^{2} \cdot 7^{2} \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 3.1.3332.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.68.2t1.a.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.3332.1

## Defining polynomial

 $f(x)$ $=$ $$x^{3} - x^{2} + 12x - 20$$ x^3 - x^2 + 12*x - 20 .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$9 + 2\cdot 23 + 5\cdot 23^{2} + 15\cdot 23^{3} + 18\cdot 23^{4} +O(23^{5})$$ 9 + 2*23 + 5*23^2 + 15*23^3 + 18*23^4+O(23^5) $r_{ 2 }$ $=$ $$18 + 23 + 2\cdot 23^{2} + 9\cdot 23^{3} + 20\cdot 23^{4} +O(23^{5})$$ 18 + 23 + 2*23^2 + 9*23^3 + 20*23^4+O(23^5) $r_{ 3 }$ $=$ $$20 + 18\cdot 23 + 15\cdot 23^{2} + 21\cdot 23^{3} + 6\cdot 23^{4} +O(23^{5})$$ 20 + 18*23 + 15*23^2 + 21*23^3 + 6*23^4+O(23^5)

## 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.