# Properties

 Label 3.751.4t5.1c1 Dimension 3 Group $S_4$ Conductor $751$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $751$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + x^{2} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Odd Determinant: 1.751.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $32 + 98\cdot 163 + 133\cdot 163^{2} + 94\cdot 163^{3} + 70\cdot 163^{4} +O\left(163^{ 5 }\right)$ $r_{ 2 }$ $=$ $34 + 7\cdot 163 + 75\cdot 163^{2} + 5\cdot 163^{3} + 99\cdot 163^{4} +O\left(163^{ 5 }\right)$ $r_{ 3 }$ $=$ $118 + 112\cdot 163 + 142\cdot 163^{2} + 115\cdot 163^{3} + 75\cdot 163^{4} +O\left(163^{ 5 }\right)$ $r_{ 4 }$ $=$ $144 + 107\cdot 163 + 137\cdot 163^{2} + 109\cdot 163^{3} + 80\cdot 163^{4} +O\left(163^{ 5 }\right)$

### 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.