# Properties

 Label 1.39.4t1.a.b Dimension $1$ Group $C_4$ Conductor $39$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$39$$$$\medspace = 3 \cdot 13$$ Artin field: Galois closure of 4.4.19773.1 Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{39}(8,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} - 11x^{2} - 9x + 3$$ x^4 - x^3 - 11*x^2 - 9*x + 3 .

The roots of $f$ are computed in $\Q_{ 17 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$1 + 13\cdot 17 + 8\cdot 17^{2} + 16\cdot 17^{3} +O(17^{5})$$ 1 + 13*17 + 8*17^2 + 16*17^3+O(17^5) $r_{ 2 }$ $=$ $$2 + 3\cdot 17 + 11\cdot 17^{2} + 13\cdot 17^{3} + 5\cdot 17^{4} +O(17^{5})$$ 2 + 3*17 + 11*17^2 + 13*17^3 + 5*17^4+O(17^5) $r_{ 3 }$ $=$ $$4 + 2\cdot 17 + 2\cdot 17^{2} + 9\cdot 17^{3} + 4\cdot 17^{4} +O(17^{5})$$ 4 + 2*17 + 2*17^2 + 9*17^3 + 4*17^4+O(17^5) $r_{ 4 }$ $=$ $$11 + 15\cdot 17 + 11\cdot 17^{2} + 11\cdot 17^{3} + 5\cdot 17^{4} +O(17^{5})$$ 11 + 15*17 + 11*17^2 + 11*17^3 + 5*17^4+O(17^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.