Properties

 Label 2.612.4t3.b Dimension $2$ Group $D_{4}$ Conductor $612$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$612$$$$\medspace = 2^{2} \cdot 3^{2} \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.0.2448.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{17})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 89 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$47 + 67\cdot 89 + 34\cdot 89^{2} + 6\cdot 89^{3} + 57\cdot 89^{4} +O(89^{5})$$ 47 + 67*89 + 34*89^2 + 6*89^3 + 57*89^4+O(89^5) $r_{ 2 }$ $=$ $$66 + 39\cdot 89 + 4\cdot 89^{2} + 86\cdot 89^{3} + 75\cdot 89^{4} +O(89^{5})$$ 66 + 39*89 + 4*89^2 + 86*89^3 + 75*89^4+O(89^5) $r_{ 3 }$ $=$ $$76 + 64\cdot 89 + 8\cdot 89^{2} + 54\cdot 89^{3} + 86\cdot 89^{4} +O(89^{5})$$ 76 + 64*89 + 8*89^2 + 54*89^3 + 86*89^4+O(89^5) $r_{ 4 }$ $=$ $$78 + 5\cdot 89 + 41\cdot 89^{2} + 31\cdot 89^{3} + 47\cdot 89^{4} +O(89^{5})$$ 78 + 5*89 + 41*89^2 + 31*89^3 + 47*89^4+O(89^5)

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

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

Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character values $c1$ $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,4)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,3)$ $0$ $2$ $4$ $(1,4,3,2)$ $0$
The blue line marks the conjugacy class containing complex conjugation.