# Properties

 Label 2.3743.4t3.a.a Dimension $2$ Group $D_{4}$ Conductor $3743$ Root number $1$ Indicator $1$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $D_{4}$ Conductor: $$3743$$$$\medspace = 19 \cdot 197$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: 4.0.71117.1 Galois orbit size: $1$ Smallest permutation container: $D_{4}$ Parity: odd Determinant: 1.3743.2t1.a.a Projective image: $C_2^2$ Projective field: $$\Q(\sqrt{-19}, \sqrt{197})$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 11$$  .

The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$8 + 45\cdot 47 + 27\cdot 47^{2} + 9\cdot 47^{3} + 43\cdot 47^{4} +O(47^{5})$$ $r_{ 2 }$ $=$ $$10 + 41\cdot 47 + 17\cdot 47^{2} + 34\cdot 47^{3} + 32\cdot 47^{4} +O(47^{5})$$ $r_{ 3 }$ $=$ $$38 + 5\cdot 47 + 29\cdot 47^{2} + 12\cdot 47^{3} + 14\cdot 47^{4} +O(47^{5})$$ $r_{ 4 }$ $=$ $$40 + 47 + 19\cdot 47^{2} + 37\cdot 47^{3} + 3\cdot 47^{4} +O(47^{5})$$

## 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.