# Properties

 Label 1.41.5t1.a.a Dimension $1$ Group $C_5$ Conductor $41$ Root number not computed Indicator $0$

# Learn more about

## Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $$41$$ Artin number field: Galois closure of 5.5.2825761.1 Galois orbit size: $4$ Smallest permutation container: $C_5$ Parity: even Dirichlet character: $$\chi_{41}(16,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{5} - x^{4} - 16 x^{3} - 5 x^{2} + 21 x + 9$.

The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $41 + 65\cdot 73 + 54\cdot 73^{2} + 73^{3} + 5\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 2 }$ $=$ $53 + 28\cdot 73 + 29\cdot 73^{2} + 32\cdot 73^{3} + 6\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 3 }$ $=$ $64 + 69\cdot 73 + 57\cdot 73^{2} + 38\cdot 73^{3} + 42\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 4 }$ $=$ $67 + 26\cdot 73 + 36\cdot 73^{2} + 59\cdot 73^{3} + 47\cdot 73^{4} +O\left(73^{ 5 }\right)$ $r_{ 5 }$ $=$ $68 + 27\cdot 73 + 40\cdot 73^{2} + 13\cdot 73^{3} + 44\cdot 73^{4} +O\left(73^{ 5 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.