# Properties

 Label 2.136.4t3.b.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 $f= x^{4} - 2 x^{3} + x^{2} - 2 x + 3$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Odd Determinant: 1.136.2t1.b.a

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $25 + 16\cdot 43 + 8\cdot 43^{2} + 19\cdot 43^{3} + 9\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 2 }$ $=$ $30 + 27\cdot 43 + 31\cdot 43^{2} + 12\cdot 43^{3} + 35\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 3 }$ $=$ $35 + 7\cdot 43 + 3\cdot 43^{2} + 31\cdot 43^{3} + 41\cdot 43^{4} +O\left(43^{ 5 }\right)$ $r_{ 4 }$ $=$ $41 + 33\cdot 43 + 42\cdot 43^{2} + 22\cdot 43^{3} + 42\cdot 43^{4} +O\left(43^{ 5 }\right)$

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

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

### 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.