Properties

 Label 2.2304.4t3.f Dimension $2$ Group $D_{4}$ Conductor $2304$ Indicator $1$

Related objects

Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$2304$$$$\medspace = 2^{8} \cdot 3^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.2.55296.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{6})$$

Galois action

Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 5 }$ to precision 7.
Roots:
 $r_{ 1 }$ $=$ $$1 + 4\cdot 5 + 4\cdot 5^{2} + 5^{3} + 3\cdot 5^{4} + 5^{5} + 3\cdot 5^{6} +O(5^{7})$$ 1 + 4*5 + 4*5^2 + 5^3 + 3*5^4 + 5^5 + 3*5^6+O(5^7) $r_{ 2 }$ $=$ $$2 + 4\cdot 5 + 3\cdot 5^{3} + 2\cdot 5^{5} +O(5^{7})$$ 2 + 4*5 + 3*5^3 + 2*5^5+O(5^7) $r_{ 3 }$ $=$ $$3 + 4\cdot 5^{2} + 5^{3} + 4\cdot 5^{4} + 2\cdot 5^{5} + 4\cdot 5^{6} +O(5^{7})$$ 3 + 4*5^2 + 5^3 + 4*5^4 + 2*5^5 + 4*5^6+O(5^7) $r_{ 4 }$ $=$ $$4 + 3\cdot 5^{3} + 5^{4} + 3\cdot 5^{5} + 5^{6} +O(5^{7})$$ 4 + 3*5^3 + 5^4 + 3*5^5 + 5^6+O(5^7)

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.