Properties

 Label 2.171.4t3.c.a Dimension $2$ Group $D_{4}$ Conductor $171$ Root number $1$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$171$$$$\medspace = 3^{2} \cdot 19$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.513.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.19.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{-19})$$

Defining polynomial

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

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$18 + 2\cdot 43 + 23\cdot 43^{2} + 25\cdot 43^{3} + 35\cdot 43^{4} +O(43^{5})$$ 18 + 2*43 + 23*43^2 + 25*43^3 + 35*43^4+O(43^5) $r_{ 2 }$ $=$ $$19 + 7\cdot 43 + 43^{2} + 38\cdot 43^{3} + 13\cdot 43^{4} +O(43^{5})$$ 19 + 7*43 + 43^2 + 38*43^3 + 13*43^4+O(43^5) $r_{ 3 }$ $=$ $$20 + 12\cdot 43 + 7\cdot 43^{2} + 39\cdot 43^{3} + 2\cdot 43^{4} +O(43^{5})$$ 20 + 12*43 + 7*43^2 + 39*43^3 + 2*43^4+O(43^5) $r_{ 4 }$ $=$ $$30 + 20\cdot 43 + 11\cdot 43^{2} + 26\cdot 43^{3} + 33\cdot 43^{4} +O(43^{5})$$ 30 + 20*43 + 11*43^2 + 26*43^3 + 33*43^4+O(43^5)

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,2)(3,4)$ $-2$ $2$ $2$ $(1,3)(2,4)$ $0$ $2$ $2$ $(1,2)$ $0$ $2$ $4$ $(1,4,2,3)$ $0$

The blue line marks the conjugacy class containing complex conjugation.