# Properties

 Label 2.5_31.4t3.1c1 Dimension 2 Group $D_{4}$ Conductor $5 \cdot 31$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $155= 5 \cdot 31$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 3 x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.5_31.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $14 + 13\cdot 41 + 3\cdot 41^{2} + 10\cdot 41^{3} + 25\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $17 + 13\cdot 41 + 14\cdot 41^{2} + 29\cdot 41^{3} + 29\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $21 + 5\cdot 41 + 18\cdot 41^{2} + 20\cdot 41^{3} + 28\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $31 + 8\cdot 41 + 5\cdot 41^{2} + 22\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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