Properties

 Label 2.39.4t3.a Dimension $2$ Group $D_{4}$ Conductor $39$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$39$$$$\medspace = 3 \cdot 13$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.117.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{13})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$11 + 27\cdot 43 + 43^{2} + 19\cdot 43^{3} + 24\cdot 43^{4} +O(43^{5})$$ 11 + 27*43 + 43^2 + 19*43^3 + 24*43^4+O(43^5) $r_{ 2 }$ $=$ $$12 + 11\cdot 43 + 32\cdot 43^{2} + 35\cdot 43^{3} + 18\cdot 43^{4} +O(43^{5})$$ 12 + 11*43 + 32*43^2 + 35*43^3 + 18*43^4+O(43^5) $r_{ 3 }$ $=$ $$25 + 41\cdot 43 + 34\cdot 43^{2} + 27\cdot 43^{3} + 30\cdot 43^{4} +O(43^{5})$$ 25 + 41*43 + 34*43^2 + 27*43^3 + 30*43^4+O(43^5) $r_{ 4 }$ $=$ $$39 + 5\cdot 43 + 17\cdot 43^{2} + 3\cdot 43^{3} + 12\cdot 43^{4} +O(43^{5})$$ 39 + 5*43 + 17*43^2 + 3*43^3 + 12*43^4+O(43^5)

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 values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.