Properties

 Label 2.152.3t2.b.a Dimension $2$ Group $S_3$ Conductor $152$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$152$$$$\medspace = 2^{3} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 3.1.152.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.152.2t1.b.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.152.1

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$17 + 21\cdot 37 + 7\cdot 37^{2} + 32\cdot 37^{3} + 30\cdot 37^{4} +O(37^{5})$$ 17 + 21*37 + 7*37^2 + 32*37^3 + 30*37^4+O(37^5) $r_{ 2 }$ $=$ $$25 + 4\cdot 37 + 31\cdot 37^{2} + 11\cdot 37^{3} + 2\cdot 37^{4} +O(37^{5})$$ 25 + 4*37 + 31*37^2 + 11*37^3 + 2*37^4+O(37^5) $r_{ 3 }$ $=$ $$33 + 10\cdot 37 + 35\cdot 37^{2} + 29\cdot 37^{3} + 3\cdot 37^{4} +O(37^{5})$$ 33 + 10*37 + 35*37^2 + 29*37^3 + 3*37^4+O(37^5)

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.