# Properties

 Label 2.2248.4t3.d.a Dimension $2$ Group $D_{4}$ Conductor $2248$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$2248$$$$\medspace = 2^{3} \cdot 281$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.2.631688.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.2248.2t1.b.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-2}, \sqrt{281})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} + 9x^{2} - 50$$ x^4 + 9*x^2 - 50 .

The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$25 + 80\cdot 283 + 155\cdot 283^{2} + 69\cdot 283^{3} + 145\cdot 283^{4} +O(283^{5})$$ 25 + 80*283 + 155*283^2 + 69*283^3 + 145*283^4+O(283^5) $r_{ 2 }$ $=$ $$82 + 255\cdot 283 + 46\cdot 283^{2} + 254\cdot 283^{3} + 259\cdot 283^{4} +O(283^{5})$$ 82 + 255*283 + 46*283^2 + 254*283^3 + 259*283^4+O(283^5) $r_{ 3 }$ $=$ $$201 + 27\cdot 283 + 236\cdot 283^{2} + 28\cdot 283^{3} + 23\cdot 283^{4} +O(283^{5})$$ 201 + 27*283 + 236*283^2 + 28*283^3 + 23*283^4+O(283^5) $r_{ 4 }$ $=$ $$258 + 202\cdot 283 + 127\cdot 283^{2} + 213\cdot 283^{3} + 137\cdot 283^{4} +O(283^{5})$$ 258 + 202*283 + 127*283^2 + 213*283^3 + 137*283^4+O(283^5)

## 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 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.