# Properties

 Label 2.95.4t3.c.a Dimension 2 Group $D_{4}$ Conductor $5 \cdot 19$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $95= 5 \cdot 19$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 2 x^{2} - x - 1$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.95.2t1.a.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 131 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $35 + 56\cdot 131 + 56\cdot 131^{2} + 104\cdot 131^{3} +O\left(131^{ 5 }\right)$ $r_{ 2 }$ $=$ $40 + 47\cdot 131 + 85\cdot 131^{2} + 72\cdot 131^{3} + 46\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 3 }$ $=$ $92 + 83\cdot 131 + 45\cdot 131^{2} + 58\cdot 131^{3} + 84\cdot 131^{4} +O\left(131^{ 5 }\right)$ $r_{ 4 }$ $=$ $97 + 74\cdot 131 + 74\cdot 131^{2} + 26\cdot 131^{3} + 130\cdot 131^{4} +O\left(131^{ 5 }\right)$

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

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

### Character values on conjugacy classes

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