# Properties

 Label 2.1129.3t2.a.a Dimension $2$ Group $S_3$ Conductor $1129$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$1129$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.3.1129.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: even Determinant: 1.1129.2t1.a.a Projective image: $S_3$ Projective field: Galois closure of 3.3.1129.1

## Defining polynomial

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

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

Roots:
 $r_{ 1 }$ $=$ $7 + 27\cdot 97 + 4\cdot 97^{2} + 33\cdot 97^{3} + 71\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 2 }$ $=$ $20 + 34\cdot 97 + 69\cdot 97^{2} + 28\cdot 97^{3} + 22\cdot 97^{4} +O\left(97^{ 5 }\right)$ $r_{ 3 }$ $=$ $70 + 35\cdot 97 + 23\cdot 97^{2} + 35\cdot 97^{3} + 3\cdot 97^{4} +O\left(97^{ 5 }\right)$

## 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.