# Properties

 Label 1.799.4t1.a.a Dimension $1$ Group $C_4$ Conductor $799$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$799$$$$\medspace = 17 \cdot 47$$ Artin field: Galois closure of 4.0.10852817.2 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: odd Dirichlet character: $$\chi_{799}(234,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} + 198x^{2} + x + 9589$$ x^4 - x^3 + 198*x^2 + x + 9589 .

The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$6 + 3\cdot 19 + 2\cdot 19^{2} + 5\cdot 19^{3} + 10\cdot 19^{4} +O(19^{5})$$ 6 + 3*19 + 2*19^2 + 5*19^3 + 10*19^4+O(19^5) $r_{ 2 }$ $=$ $$7 + 2\cdot 19 + 18\cdot 19^{2} + 14\cdot 19^{3} + 9\cdot 19^{4} +O(19^{5})$$ 7 + 2*19 + 18*19^2 + 14*19^3 + 9*19^4+O(19^5) $r_{ 3 }$ $=$ $$10 + 3\cdot 19 + 14\cdot 19^{2} + 11\cdot 19^{3} + 3\cdot 19^{4} +O(19^{5})$$ 10 + 3*19 + 14*19^2 + 11*19^3 + 3*19^4+O(19^5) $r_{ 4 }$ $=$ $$16 + 9\cdot 19 + 3\cdot 19^{2} + 6\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})$$ 16 + 9*19 + 3*19^2 + 6*19^3 + 14*19^4+O(19^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.