Properties

 Label 3.3267.4t5.c.a Dimension $3$ Group $S_4$ Conductor $3267$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $$3267$$$$\medspace = 3^{3} \cdot 11^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.2.3267.1 Galois orbit size: $1$ Smallest permutation container: $S_4$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $S_4$ Projective stem field: Galois closure of 4.2.3267.1

Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + 3x^{2} + x - 1$$ x^4 - x^3 + 3*x^2 + x - 1 .

The roots of $f$ are computed in $\Q_{ 193 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$5 + 113\cdot 193 + 4\cdot 193^{2} + 59\cdot 193^{3} + 153\cdot 193^{4} +O(193^{5})$$ 5 + 113*193 + 4*193^2 + 59*193^3 + 153*193^4+O(193^5) $r_{ 2 }$ $=$ $$43 + 138\cdot 193 + 174\cdot 193^{2} + 118\cdot 193^{3} + 143\cdot 193^{4} +O(193^{5})$$ 43 + 138*193 + 174*193^2 + 118*193^3 + 143*193^4+O(193^5) $r_{ 3 }$ $=$ $$71 + 20\cdot 193 + 120\cdot 193^{2} + 167\cdot 193^{3} + 65\cdot 193^{4} +O(193^{5})$$ 71 + 20*193 + 120*193^2 + 167*193^3 + 65*193^4+O(193^5) $r_{ 4 }$ $=$ $$75 + 114\cdot 193 + 86\cdot 193^{2} + 40\cdot 193^{3} + 23\cdot 193^{4} +O(193^{5})$$ 75 + 114*193 + 86*193^2 + 40*193^3 + 23*193^4+O(193^5)

Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,2,3,4)$ $(1,2)$

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $6$ $2$ $(1,2)$ $1$ $8$ $3$ $(1,2,3)$ $0$ $6$ $4$ $(1,2,3,4)$ $-1$

The blue line marks the conjugacy class containing complex conjugation.