# Properties

 Label 2.5952.4t3.e.a Dimension $2$ Group $D_{4}$ Conductor $5952$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$5952$$$$\medspace = 2^{6} \cdot 3 \cdot 31$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 4.2.738048.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.372.2t1.a.a Projective image: $C_2^2$ Projective field: Galois closure of $$\Q(\sqrt{-3}, \sqrt{31})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 4x^{2} - 27$$ x^4 - 4*x^2 - 27 .

The roots of $f$ are computed in $\Q_{ 79 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$29 + 55\cdot 79 + 60\cdot 79^{2} + 44\cdot 79^{3} + 9\cdot 79^{4} +O(79^{5})$$ 29 + 55*79 + 60*79^2 + 44*79^3 + 9*79^4+O(79^5) $r_{ 2 }$ $=$ $$36 + 42\cdot 79 + 64\cdot 79^{2} + 11\cdot 79^{3} + 73\cdot 79^{4} +O(79^{5})$$ 36 + 42*79 + 64*79^2 + 11*79^3 + 73*79^4+O(79^5) $r_{ 3 }$ $=$ $$43 + 36\cdot 79 + 14\cdot 79^{2} + 67\cdot 79^{3} + 5\cdot 79^{4} +O(79^{5})$$ 43 + 36*79 + 14*79^2 + 67*79^3 + 5*79^4+O(79^5) $r_{ 4 }$ $=$ $$50 + 23\cdot 79 + 18\cdot 79^{2} + 34\cdot 79^{3} + 69\cdot 79^{4} +O(79^{5})$$ 50 + 23*79 + 18*79^2 + 34*79^3 + 69*79^4+O(79^5)

## 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 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.