# Properties

 Label 2.136.4t3.c.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 4.0.2312.1 defined by $f= x^{4} - x^{3} - 2 x + 4$ over $\Q$ Size of Galois orbit: 1 Smallest containing permutation representation: $D_{4}$ Parity: Even Determinant: 1.136.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{17})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $4 + 5\cdot 47 + 44\cdot 47^{2} + 20\cdot 47^{3} + 22\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $8 + 21\cdot 47 + 10\cdot 47^{2} + 9\cdot 47^{3} + 38\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $12 + 27\cdot 47 + 47^{2} + 7\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $24 + 40\cdot 47 + 37\cdot 47^{2} + 9\cdot 47^{3} + 12\cdot 47^{4} +O\left(47^{ 5 }\right)$

### 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.