# Properties

 Label 1.25.5t1.a.a Dimension $1$ Group $C_5$ Conductor $25$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_5$ Conductor: $$25$$$$\medspace = 5^{2}$$ Artin field: 5.5.390625.1 Galois orbit size: $4$ Smallest permutation container: $C_5$ Parity: even Dirichlet character: $$\chi_{25}(11,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{5} - 10 x^{3} - 5 x^{2} + 10 x - 1$$  .

The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.

Roots:
 $r_{ 1 }$ $=$ $$3 + 3\cdot 43 + 24\cdot 43^{2} + 5\cdot 43^{3} + 7\cdot 43^{4} +O(43^{5})$$ $r_{ 2 }$ $=$ $$7 + 13\cdot 43 + 19\cdot 43^{2} + 5\cdot 43^{3} + 21\cdot 43^{4} +O(43^{5})$$ $r_{ 3 }$ $=$ $$19 + 14\cdot 43 + 26\cdot 43^{2} + 32\cdot 43^{3} + 5\cdot 43^{4} +O(43^{5})$$ $r_{ 4 }$ $=$ $$24 + 12\cdot 43 + 2\cdot 43^{2} + 38\cdot 43^{3} + 26\cdot 43^{4} +O(43^{5})$$ $r_{ 5 }$ $=$ $$33 + 42\cdot 43 + 13\cdot 43^{2} + 4\cdot 43^{3} + 25\cdot 43^{4} +O(43^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.