# Properties

 Label 2.400.4t3.b Dimension $2$ Group $D_{4}$ Conductor $400$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$400$$$$\medspace = 2^{4} \cdot 5^{2}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin number field: Galois closure of 4.2.2000.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Projective image: $C_2^2$ Projective field: $$\Q(i, \sqrt{5})$$

## Galois action

### Roots of defining polynomial

The roots of $f$ are computed in $\Q_{ 101 }$ to precision 5.
Roots:
 $r_{ 1 }$ $=$ $$34 + 50\cdot 101^{2} + 68\cdot 101^{3} + 34\cdot 101^{4} +O(101^{5})$$ $r_{ 2 }$ $=$ $$37 + 72\cdot 101 + 73\cdot 101^{2} + 57\cdot 101^{3} + 77\cdot 101^{4} +O(101^{5})$$ $r_{ 3 }$ $=$ $$64 + 28\cdot 101 + 27\cdot 101^{2} + 43\cdot 101^{3} + 23\cdot 101^{4} +O(101^{5})$$ $r_{ 4 }$ $=$ $$67 + 100\cdot 101 + 50\cdot 101^{2} + 32\cdot 101^{3} + 66\cdot 101^{4} +O(101^{5})$$

### 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.