# Properties

 Label 2.23.3t2.b.a Dimension $2$ Group $S_3$ Conductor $23$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$23$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 3.1.23.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Determinant: 1.23.2t1.a.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.23.1

## Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 59 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$7 + 39\cdot 59 + 39\cdot 59^{2} + 36\cdot 59^{3} + 59^{4} +O(59^{5})$$ 7 + 39*59 + 39*59^2 + 36*59^3 + 59^4+O(59^5) $r_{ 2 }$ $=$ $$9 + 59 + 51\cdot 59^{2} + 39\cdot 59^{3} + 51\cdot 59^{4} +O(59^{5})$$ 9 + 59 + 51*59^2 + 39*59^3 + 51*59^4+O(59^5) $r_{ 3 }$ $=$ $$44 + 18\cdot 59 + 27\cdot 59^{2} + 41\cdot 59^{3} + 5\cdot 59^{4} +O(59^{5})$$ 44 + 18*59 + 27*59^2 + 41*59^3 + 5*59^4+O(59^5)

## 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.

This representation has the smallest conductor of all $2$-dimensional irreducible faithful Artin representations.