Properties

 Label 2.768.4t3.b.a Dimension $2$ Group $D_{4}$ Conductor $768$ Root number $1$ Indicator $1$

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Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$768$$$$\medspace = 2^{8} \cdot 3$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.2.18432.2 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.3.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{-3})$$

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$15 + 23\cdot 67 + 30\cdot 67^{2} + 43\cdot 67^{3} + 7\cdot 67^{4} +O(67^{5})$$ 15 + 23*67 + 30*67^2 + 43*67^3 + 7*67^4+O(67^5) $r_{ 2 }$ $=$ $$21 + 28\cdot 67 + 28\cdot 67^{2} + 59\cdot 67^{3} + 7\cdot 67^{4} +O(67^{5})$$ 21 + 28*67 + 28*67^2 + 59*67^3 + 7*67^4+O(67^5) $r_{ 3 }$ $=$ $$46 + 38\cdot 67 + 38\cdot 67^{2} + 7\cdot 67^{3} + 59\cdot 67^{4} +O(67^{5})$$ 46 + 38*67 + 38*67^2 + 7*67^3 + 59*67^4+O(67^5) $r_{ 4 }$ $=$ $$52 + 43\cdot 67 + 36\cdot 67^{2} + 23\cdot 67^{3} + 59\cdot 67^{4} +O(67^{5})$$ 52 + 43*67 + 36*67^2 + 23*67^3 + 59*67^4+O(67^5)

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.