# Properties

 Label 1.571.5t1.a.b Dimension $1$ Group $C_5$ Conductor $571$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $$571$$ Artin field: Galois closure of 5.5.106302733681.1 Galois orbit size: $4$ Smallest permutation container: $C_5$ Parity: even Dirichlet character: $$\chi_{571}(387,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} - 228x^{3} - 868x^{2} + 3056x + 7552$$ x^5 - x^4 - 228*x^3 - 868*x^2 + 3056*x + 7552 .

The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$1 + 9\cdot 29 + 14\cdot 29^{2} + 16\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})$$ 1 + 9*29 + 14*29^2 + 16*29^3 + 16*29^4+O(29^5) $r_{ 2 }$ $=$ $$17 + 3\cdot 29 + 18\cdot 29^{2} + 10\cdot 29^{3} + 20\cdot 29^{4} +O(29^{5})$$ 17 + 3*29 + 18*29^2 + 10*29^3 + 20*29^4+O(29^5) $r_{ 3 }$ $=$ $$21 + 29 + 24\cdot 29^{2} + 9\cdot 29^{3} + 21\cdot 29^{4} +O(29^{5})$$ 21 + 29 + 24*29^2 + 9*29^3 + 21*29^4+O(29^5) $r_{ 4 }$ $=$ $$23 + 8\cdot 29 + 4\cdot 29^{2} + 12\cdot 29^{4} +O(29^{5})$$ 23 + 8*29 + 4*29^2 + 12*29^4+O(29^5) $r_{ 5 }$ $=$ $$26 + 5\cdot 29 + 26\cdot 29^{2} + 20\cdot 29^{3} + 16\cdot 29^{4} +O(29^{5})$$ 26 + 5*29 + 26*29^2 + 20*29^3 + 16*29^4+O(29^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 5 }$

 Cycle notation $(1,3,5,2,4)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 5 }$ Character value $1$ $1$ $()$ $1$ $1$ $5$ $(1,3,5,2,4)$ $\zeta_{5}^{2}$ $1$ $5$ $(1,5,4,3,2)$ $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ $1$ $5$ $(1,2,3,4,5)$ $\zeta_{5}$ $1$ $5$ $(1,4,2,5,3)$ $\zeta_{5}^{3}$

The blue line marks the conjugacy class containing complex conjugation.