# Properties

 Label 3.2e3_127.4t5.1c1 Dimension 3 Group $S_4$ Conductor $2^{3} \cdot 127$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $1016= 2^{3} \cdot 127$ Artin number field: Splitting field of $f= x^{4} - x^{3} + x^{2} - 2 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Even Determinant: 1.2e3_127.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 173 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $48 + 75\cdot 173 + 91\cdot 173^{2} + 119\cdot 173^{3} + 156\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 2 }$ $=$ $57 + 15\cdot 173 + 94\cdot 173^{2} + 64\cdot 173^{3} + 143\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 3 }$ $=$ $73 + 28\cdot 173 + 88\cdot 173^{2} + 171\cdot 173^{3} + 2\cdot 173^{4} +O\left(173^{ 5 }\right)$ $r_{ 4 }$ $=$ $169 + 53\cdot 173 + 72\cdot 173^{2} + 163\cdot 173^{3} + 42\cdot 173^{4} +O\left(173^{ 5 }\right)$

### Generators of the action on the roots $r_1, \ldots, r_{ 4 }$

 Cycle notation $(1,2,3,4)$ $(1,2)$

### Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $3$ $3$ $2$ $(1,2)(3,4)$ $-1$ $6$ $2$ $(1,2)$ $1$ $8$ $3$ $(1,2,3)$ $0$ $6$ $4$ $(1,2,3,4)$ $-1$
The blue line marks the conjugacy class containing complex conjugation.