# Properties

 Label 2.1596.4t3.l.a Dimension $2$ Group $D_{4}$ Conductor $1596$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$1596$$$$\medspace = 2^{2} \cdot 3 \cdot 7 \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.44688.2 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: even Determinant: 1.1596.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{7}, \sqrt{57})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 2x^{3} + 6x^{2} + 2x + 29$$ x^4 - 2*x^3 + 6*x^2 + 2*x + 29 .

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$14\cdot 29 + 6\cdot 29^{2} + 29^{3} + 24\cdot 29^{4} +O(29^{5})$$ 14*29 + 6*29^2 + 29^3 + 24*29^4+O(29^5) $r_{ 2 }$ $=$ $$3 + 5\cdot 29 + 13\cdot 29^{2} + 17\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})$$ 3 + 5*29 + 13*29^2 + 17*29^3 + 2*29^4+O(29^5) $r_{ 3 }$ $=$ $$4 + 7\cdot 29 + 6\cdot 29^{2} + 14\cdot 29^{3} + 5\cdot 29^{4} +O(29^{5})$$ 4 + 7*29 + 6*29^2 + 14*29^3 + 5*29^4+O(29^5) $r_{ 4 }$ $=$ $$24 + 2\cdot 29 + 3\cdot 29^{2} + 25\cdot 29^{3} + 25\cdot 29^{4} +O(29^{5})$$ 24 + 2*29 + 3*29^2 + 25*29^3 + 25*29^4+O(29^5)

## 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.