# Properties

 Label 3.2e2_359.4t5.1c1 Dimension 3 Group $S_4$ Conductor $2^{2} \cdot 359$ Root number 1 Frobenius-Schur indicator 1

# Learn more about

## Basic invariants

 Dimension: $3$ Group: $S_4$ Conductor: $1436= 2^{2} \cdot 359$ Artin number field: Splitting field of $f= x^{4} - x^{3} + 3 x^{2} + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $S_4$ Parity: Even Determinant: 1.2e2_359.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 103 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $9 + 3\cdot 103 + 6\cdot 103^{2} + 45\cdot 103^{3} + 5\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 2 }$ $=$ $25 + 51\cdot 103 + 22\cdot 103^{2} + 10\cdot 103^{3} + 21\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 3 }$ $=$ $31 + 6\cdot 103 + 14\cdot 103^{2} + 97\cdot 103^{3} + 49\cdot 103^{4} +O\left(103^{ 5 }\right)$ $r_{ 4 }$ $=$ $39 + 42\cdot 103 + 60\cdot 103^{2} + 53\cdot 103^{3} + 26\cdot 103^{4} +O\left(103^{ 5 }\right)$

### 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.