# Properties

 Label 2.1556.3t2.a.a Dimension 2 Group $S_3$ Conductor $2^{2} \cdot 389$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $1556= 2^{2} \cdot 389$ Artin number field: Splitting field of 3.3.1556.1 defined by $f= x^{3} - x^{2} - 9 x + 11$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Even Determinant: 1.389.2t1.a.a Projective image: $S_3$ Projective field: Galois closure of 3.3.1556.1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $5\cdot 11 + 9\cdot 11^{2} + 4\cdot 11^{3} + 4\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 2 }$ $=$ $5 + 5\cdot 11 + 9\cdot 11^{2} + 9\cdot 11^{3} + 8\cdot 11^{4} +O\left(11^{ 5 }\right)$ $r_{ 3 }$ $=$ $7 + 3\cdot 11^{2} + 7\cdot 11^{3} + 8\cdot 11^{4} +O\left(11^{ 5 }\right)$

### 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.