# Properties

 Label 2.776.4t3.b.a Dimension $2$ Group $D_{4}$ Conductor $776$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$776$$$$\medspace = 2^{3} \cdot 97$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.0.75272.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: even Determinant: 1.776.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{2}, \sqrt{97})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 13x^{2} + 18$$ x^4 + 13*x^2 + 18 .

The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.

Roots:
 $r_{ 1 }$ $=$ $$14 + 24\cdot 31 + 5\cdot 31^{2} + 2\cdot 31^{3} + 28\cdot 31^{4} + 2\cdot 31^{5} +O(31^{6})$$ 14 + 24*31 + 5*31^2 + 2*31^3 + 28*31^4 + 2*31^5+O(31^6) $r_{ 2 }$ $=$ $$15 + 4\cdot 31 + 14\cdot 31^{2} + 12\cdot 31^{3} + 14\cdot 31^{4} + 30\cdot 31^{5} +O(31^{6})$$ 15 + 4*31 + 14*31^2 + 12*31^3 + 14*31^4 + 30*31^5+O(31^6) $r_{ 3 }$ $=$ $$16 + 26\cdot 31 + 16\cdot 31^{2} + 18\cdot 31^{3} + 16\cdot 31^{4} +O(31^{6})$$ 16 + 26*31 + 16*31^2 + 18*31^3 + 16*31^4+O(31^6) $r_{ 4 }$ $=$ $$17 + 6\cdot 31 + 25\cdot 31^{2} + 28\cdot 31^{3} + 2\cdot 31^{4} + 28\cdot 31^{5} +O(31^{6})$$ 17 + 6*31 + 25*31^2 + 28*31^3 + 2*31^4 + 28*31^5+O(31^6)

## 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 value $1$ $1$ $()$ $2$ $1$ $2$ $(1,4)(2,3)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,4)$ $0$ $2$ $4$ $(1,3,4,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.