# Properties

 Label 2.136.4t3.a.a Dimension 2 Group $D_{4}$ Conductor $2^{3} \cdot 17$ Root number 1 Frobenius-Schur indicator 1

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $136= 2^{3} \cdot 17$ Artin number field: Splitting field of $f= x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 2$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Even Determinant: 1.136.2t1.a.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $12 + 26\cdot 47 + 7\cdot 47^{2} + 30\cdot 47^{3} + 13\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $16 + 32\cdot 47 + 45\cdot 47^{2} + 27\cdot 47^{3} + 43\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $32 + 14\cdot 47 + 47^{2} + 19\cdot 47^{3} + 3\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $36 + 20\cdot 47 + 39\cdot 47^{2} + 16\cdot 47^{3} + 33\cdot 47^{4} +O\left(47^{ 5 }\right)$

### 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.