# Properties

 Label 3.8281.4t4.b.a Dimension $3$ Group $A_4$ Conductor $8281$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $3$ Group: $A_4$ Conductor: $$8281$$$$\medspace = 7^{2} \cdot 13^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 4.0.8281.1 Galois orbit size: $1$ Smallest permutation container: $A_4$ Parity: even Determinant: 1.1.1t1.a.a Projective image: $A_4$ Projective stem field: 4.0.8281.1

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + 5 x^{2} - 4 x + 3$$  .

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

Roots:
 $r_{ 1 }$ $=$ $$13 + 33\cdot 97 + 63\cdot 97^{2} + 80\cdot 97^{3} + 69\cdot 97^{4} +O(97^{5})$$ $r_{ 2 }$ $=$ $$31 + 76\cdot 97 + 57\cdot 97^{2} + 68\cdot 97^{3} + 80\cdot 97^{4} +O(97^{5})$$ $r_{ 3 }$ $=$ $$64 + 71\cdot 97 + 39\cdot 97^{2} + 9\cdot 97^{3} + 81\cdot 97^{4} +O(97^{5})$$ $r_{ 4 }$ $=$ $$87 + 12\cdot 97 + 33\cdot 97^{2} + 35\cdot 97^{3} + 59\cdot 97^{4} +O(97^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.