# Properties

 Label 2.2304.4t3.e.a Dimension $2$ Group $D_{4}$ Conductor $2304$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$2304$$$$\medspace = 2^{8} \cdot 3^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.2.55296.4 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.4.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{6})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 24$$ x^4 - 24 .

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

Roots:
 $r_{ 1 }$ $=$ $$4 + 19\cdot 29 + 28\cdot 29^{2} + 14\cdot 29^{3} + 27\cdot 29^{4} +O(29^{5})$$ 4 + 19*29 + 28*29^2 + 14*29^3 + 27*29^4+O(29^5) $r_{ 2 }$ $=$ $$10 + 27\cdot 29 + 23\cdot 29^{2} + 21\cdot 29^{3} + 2\cdot 29^{4} +O(29^{5})$$ 10 + 27*29 + 23*29^2 + 21*29^3 + 2*29^4+O(29^5) $r_{ 3 }$ $=$ $$19 + 29 + 5\cdot 29^{2} + 7\cdot 29^{3} + 26\cdot 29^{4} +O(29^{5})$$ 19 + 29 + 5*29^2 + 7*29^3 + 26*29^4+O(29^5) $r_{ 4 }$ $=$ $$25 + 9\cdot 29 + 14\cdot 29^{3} + 29^{4} +O(29^{5})$$ 25 + 9*29 + 14*29^3 + 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.