# Properties

 Label 1.15.4t1.a.b Dimension $1$ Group $C_4$ Conductor $15$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_4$ Conductor: $$15$$$$\medspace = 3 \cdot 5$$ Artin field: Galois closure of $$\Q(\zeta_{15})^+$$ Galois orbit size: $2$ Smallest permutation container: $C_4$ Parity: even Dirichlet character: $$\chi_{15}(2,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{4} - x^{3} - 4x^{2} + 4x + 1$$ x^4 - x^3 - 4*x^2 + 4*x + 1 .

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

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

## 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.