# Properties

 Label 2.2951.3t2.a Dimension $2$ Group $S_3$ Conductor $2951$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $$2951$$$$\medspace = 13 \cdot 227$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 3.1.2951.1 Galois orbit size: $1$ Smallest permutation container: $S_3$ Parity: odd Projective image: $S_3$ Projective field: Galois closure of 3.1.2951.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$1 + 3\cdot 5 + 4\cdot 5^{2} + 4\cdot 5^{4} +O(5^{5})$$ 1 + 3*5 + 4*5^2 + 4*5^4+O(5^5) $r_{ 2 }$ $=$ $$2 + 5 + 3\cdot 5^{2} + 5^{3} +O(5^{5})$$ 2 + 5 + 3*5^2 + 5^3+O(5^5) $r_{ 3 }$ $=$ $$3 + 2\cdot 5^{2} + 2\cdot 5^{3} +O(5^{5})$$ 3 + 2*5^2 + 2*5^3+O(5^5)

### 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.