# Properties

 Label 2.3600.4t3.a Dimension $2$ Group $D_{4}$ Conductor $3600$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$3600$$$$\medspace = 2^{4} \cdot 3^{2} \cdot 5^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.2.18000.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(i, \sqrt{5})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$2 + 10\cdot 29 + 18\cdot 29^{2} + 16\cdot 29^{3} + 10\cdot 29^{4} +O(29^{5})$$ 2 + 10*29 + 18*29^2 + 16*29^3 + 10*29^4+O(29^5) $r_{ 2 }$ $=$ $$5 + 22\cdot 29 + 6\cdot 29^{2} + 7\cdot 29^{3} + 25\cdot 29^{4} +O(29^{5})$$ 5 + 22*29 + 6*29^2 + 7*29^3 + 25*29^4+O(29^5) $r_{ 3 }$ $=$ $$24 + 6\cdot 29 + 22\cdot 29^{2} + 21\cdot 29^{3} + 3\cdot 29^{4} +O(29^{5})$$ 24 + 6*29 + 22*29^2 + 21*29^3 + 3*29^4+O(29^5) $r_{ 4 }$ $=$ $$27 + 18\cdot 29 + 10\cdot 29^{2} + 12\cdot 29^{3} + 18\cdot 29^{4} +O(29^{5})$$ 27 + 18*29 + 10*29^2 + 12*29^3 + 18*29^4+O(29^5)

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

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

### 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.