# Properties

 Label 2.5_11_29e2.4t3.2c1 Dimension 2 Group $D_{4}$ Conductor $5 \cdot 11 \cdot 29^{2}$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $46255= 5 \cdot 11 \cdot 29^{2}$ Artin number field: Splitting field of $f= x^{4} - x^{3} - 20 x^{2} + 50 x + 267$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.5_11.2t1.1c1

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $9 + 26\cdot 31 + 26\cdot 31^{2} + 10\cdot 31^{3} + 4\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 2 }$ $=$ $13 + 27\cdot 31 + 5\cdot 31^{2} + 20\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 3 }$ $=$ $19 + 4\cdot 31 + 11\cdot 31^{2} + 27\cdot 31^{3} + 28\cdot 31^{4} +O\left(31^{ 5 }\right)$ $r_{ 4 }$ $=$ $22 + 3\cdot 31 + 18\cdot 31^{2} + 3\cdot 31^{3} + 10\cdot 31^{4} +O\left(31^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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