# Properties

 Label 3.5031049.6t8.a.a Dimension $3$ Group $S_4$ Conductor $5031049$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $$5031049$$$$\medspace = 2243^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.2.2243.1 Galois orbit size: $1$ Smallest permutation container: $S_4$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $S_4$ Projective stem field: Galois closure of 4.2.2243.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 97 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$9 + 57\cdot 97 + 42\cdot 97^{2} + 54\cdot 97^{3} + 37\cdot 97^{4} +O(97^{5})$$ 9 + 57*97 + 42*97^2 + 54*97^3 + 37*97^4+O(97^5) $r_{ 2 }$ $=$ $$50 + 40\cdot 97 + 4\cdot 97^{2} + 76\cdot 97^{3} + 64\cdot 97^{4} +O(97^{5})$$ 50 + 40*97 + 4*97^2 + 76*97^3 + 64*97^4+O(97^5) $r_{ 3 }$ $=$ $$66 + 75\cdot 97 + 32\cdot 97^{2} + 54\cdot 97^{3} + 41\cdot 97^{4} +O(97^{5})$$ 66 + 75*97 + 32*97^2 + 54*97^3 + 41*97^4+O(97^5) $r_{ 4 }$ $=$ $$70 + 20\cdot 97 + 17\cdot 97^{2} + 9\cdot 97^{3} + 50\cdot 97^{4} +O(97^{5})$$ 70 + 20*97 + 17*97^2 + 9*97^3 + 50*97^4+O(97^5)

## 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.