# Properties

 Label 2.663.4t3.a.a Dimension $2$ Group $D_{4}$ Conductor $663$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$663$$$$\medspace = 3 \cdot 13 \cdot 17$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 4.2.8619.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.663.2t1.a.a Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{13}, \sqrt{-51})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} - 4 x^{2} - 6 x - 3$$  .

The roots of $f$ are computed in $\Q_{ 23 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$2 + 11\cdot 23 + 7\cdot 23^{2} + 18\cdot 23^{4} +O(23^{5})$$ $r_{ 2 }$ $=$ $$6 + 14\cdot 23 + 6\cdot 23^{2} + 14\cdot 23^{3} + 7\cdot 23^{4} +O(23^{5})$$ $r_{ 3 }$ $=$ $$7 + 23 + 7\cdot 23^{2} + 9\cdot 23^{3} + 16\cdot 23^{4} +O(23^{5})$$ $r_{ 4 }$ $=$ $$9 + 19\cdot 23 + 23^{2} + 22\cdot 23^{3} + 3\cdot 23^{4} +O(23^{5})$$

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 4 }$ Character value $1$ $1$ $()$ $2$ $1$ $2$ $(1,3)(2,4)$ $-2$ $2$ $2$ $(1,2)(3,4)$ $0$ $2$ $2$ $(1,3)$ $0$ $2$ $4$ $(1,4,3,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.