# Properties

 Label 2.76.3t2.a Dimension $2$ Group $S_3$ Conductor $76$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$76$$$$\medspace = 2^{2} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.76.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.76.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$12 + 4\cdot 23 + 8\cdot 23^{2} + 22\cdot 23^{3} + 10\cdot 23^{4} +O(23^{5})$$ 12 + 4*23 + 8*23^2 + 22*23^3 + 10*23^4+O(23^5) $r_{ 2 }$ $=$ $$14 + 6\cdot 23 + 5\cdot 23^{2} + 7\cdot 23^{3} + 15\cdot 23^{4} +O(23^{5})$$ 14 + 6*23 + 5*23^2 + 7*23^3 + 15*23^4+O(23^5) $r_{ 3 }$ $=$ $$20 + 11\cdot 23 + 9\cdot 23^{2} + 16\cdot 23^{3} + 19\cdot 23^{4} +O(23^{5})$$ 20 + 11*23 + 9*23^2 + 16*23^3 + 19*23^4+O(23^5)

### 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 values $c1$ $1$ $1$ $()$ $2$ $3$ $2$ $(1,2)$ $0$ $2$ $3$ $(1,2,3)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.