# Properties

 Label 2.1164.3t2.a.a Dimension $2$ Group $S_3$ Conductor $1164$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$1164$$$$\medspace = 2^{2} \cdot 3 \cdot 97$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 3.1.1164.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.291.2t1.a.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.1164.1

## Defining polynomial

 $f(x)$ $=$ $$x^{3} - x^{2} - x + 7$$ x^3 - x^2 - x + 7 .

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$4 + 13\cdot 17 + 3\cdot 17^{2} + 8\cdot 17^{3} + 7\cdot 17^{4} +O(17^{5})$$ 4 + 13*17 + 3*17^2 + 8*17^3 + 7*17^4+O(17^5) $r_{ 2 }$ $=$ $$5 + 10\cdot 17 + 11\cdot 17^{2} + 6\cdot 17^{3} + 2\cdot 17^{4} +O(17^{5})$$ 5 + 10*17 + 11*17^2 + 6*17^3 + 2*17^4+O(17^5) $r_{ 3 }$ $=$ $$9 + 10\cdot 17 + 17^{2} + 2\cdot 17^{3} + 7\cdot 17^{4} +O(17^{5})$$ 9 + 10*17 + 17^2 + 2*17^3 + 7*17^4+O(17^5)

## 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.