# Properties

 Label 2.257.3t2.a.a Dimension 2 Group $S_3$ Conductor $257$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3$ Conductor: $257$ Artin number field: Splitting field of $f= x^{3} - x^{2} - 4 x + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_3$ Parity: Even Determinant: 1.257.2t1.a.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 61 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $27 + 21\cdot 61 + 25\cdot 61^{2} + 29\cdot 61^{3} + 54\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 2 }$ $=$ $39 + 12\cdot 61 + 61^{2} + 22\cdot 61^{3} + 33\cdot 61^{4} +O\left(61^{ 5 }\right)$ $r_{ 3 }$ $=$ $57 + 26\cdot 61 + 34\cdot 61^{2} + 9\cdot 61^{3} + 34\cdot 61^{4} +O\left(61^{ 5 }\right)$

### 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.