# Properties

 Label 1.31.5t1.a.c Dimension $1$ Group $C_5$ Conductor $31$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $$31$$ Artin field: Galois closure of 5.5.923521.1 Galois orbit size: $4$ Smallest permutation container: $C_5$ Parity: even Dirichlet character: $$\chi_{31}(16,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - x^{4} - 12x^{3} + 21x^{2} + x - 5$$ x^5 - x^4 - 12*x^3 + 21*x^2 + x - 5 .

The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$10 + 20\cdot 37 + 28\cdot 37^{2} + 15\cdot 37^{3} + 5\cdot 37^{4} +O(37^{5})$$ 10 + 20*37 + 28*37^2 + 15*37^3 + 5*37^4+O(37^5) $r_{ 2 }$ $=$ $$18 + 26\cdot 37 + 8\cdot 37^{2} + 19\cdot 37^{3} + 20\cdot 37^{4} +O(37^{5})$$ 18 + 26*37 + 8*37^2 + 19*37^3 + 20*37^4+O(37^5) $r_{ 3 }$ $=$ $$20 + 8\cdot 37 + 17\cdot 37^{2} + 24\cdot 37^{3} + 4\cdot 37^{4} +O(37^{5})$$ 20 + 8*37 + 17*37^2 + 24*37^3 + 4*37^4+O(37^5) $r_{ 4 }$ $=$ $$31 + 20\cdot 37 + 10\cdot 37^{2} + 27\cdot 37^{3} + 10\cdot 37^{4} +O(37^{5})$$ 31 + 20*37 + 10*37^2 + 27*37^3 + 10*37^4+O(37^5) $r_{ 5 }$ $=$ $$33 + 34\cdot 37 + 8\cdot 37^{2} + 24\cdot 37^{3} + 32\cdot 37^{4} +O(37^{5})$$ 33 + 34*37 + 8*37^2 + 24*37^3 + 32*37^4+O(37^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.