# Properties

 Label 2.1560.4t3.d.a Dimension $2$ Group $D_{4}$ Conductor $1560$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$1560$$$$\medspace = 2^{3} \cdot 3 \cdot 5 \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.60840.4 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.1560.2t1.b.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{10}, \sqrt{-39})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 2x^{2} + 40$$ x^4 + 2*x^2 + 40 .

The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$4 + 6\cdot 41 + 11\cdot 41^{2} + 32\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})$$ 4 + 6*41 + 11*41^2 + 32*41^3 + 20*41^4+O(41^5) $r_{ 2 }$ $=$ $$8 + 2\cdot 41 + 38\cdot 41^{2} + 41^{3} + 8\cdot 41^{4} +O(41^{5})$$ 8 + 2*41 + 38*41^2 + 41^3 + 8*41^4+O(41^5) $r_{ 3 }$ $=$ $$33 + 38\cdot 41 + 2\cdot 41^{2} + 39\cdot 41^{3} + 32\cdot 41^{4} +O(41^{5})$$ 33 + 38*41 + 2*41^2 + 39*41^3 + 32*41^4+O(41^5) $r_{ 4 }$ $=$ $$37 + 34\cdot 41 + 29\cdot 41^{2} + 8\cdot 41^{3} + 20\cdot 41^{4} +O(41^{5})$$ 37 + 34*41 + 29*41^2 + 8*41^3 + 20*41^4+O(41^5)

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

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

## 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.