# Properties

 Label 1.760.6t1.b.a Dimension $1$ Group $C_6$ Conductor $760$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$760$$$$\medspace = 2^{3} \cdot 5 \cdot 19$$ Artin field: Galois closure of 6.0.8340544000.3 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{760}(619,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} + 19x^{4} - 14x^{3} + 342x^{2} - 584x + 2849$$ x^6 - 2*x^5 + 19*x^4 - 14*x^3 + 342*x^2 - 584*x + 2849 .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{2} + 29x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$6 a + 29 + \left(a + 11\right)\cdot 31 + \left(a + 27\right)\cdot 31^{2} + \left(11 a + 20\right)\cdot 31^{3} + \left(21 a + 19\right)\cdot 31^{4} +O(31^{5})$$ 6*a + 29 + (a + 11)*31 + (a + 27)*31^2 + (11*a + 20)*31^3 + (21*a + 19)*31^4+O(31^5) $r_{ 2 }$ $=$ $$25 a + 22 + 29 a\cdot 31 + \left(29 a + 22\right)\cdot 31^{2} + \left(19 a + 11\right)\cdot 31^{3} + \left(9 a + 9\right)\cdot 31^{4} +O(31^{5})$$ 25*a + 22 + 29*a*31 + (29*a + 22)*31^2 + (19*a + 11)*31^3 + (9*a + 9)*31^4+O(31^5) $r_{ 3 }$ $=$ $$6 a + 6 + \left(a + 20\right)\cdot 31 + \left(a + 27\right)\cdot 31^{2} + \left(11 a + 18\right)\cdot 31^{3} + \left(21 a + 1\right)\cdot 31^{4} +O(31^{5})$$ 6*a + 6 + (a + 20)*31 + (a + 27)*31^2 + (11*a + 18)*31^3 + (21*a + 1)*31^4+O(31^5) $r_{ 4 }$ $=$ $$6 a + 10 + \left(a + 4\right)\cdot 31 + \left(a + 21\right)\cdot 31^{2} + \left(11 a + 21\right)\cdot 31^{3} + \left(21 a + 8\right)\cdot 31^{4} +O(31^{5})$$ 6*a + 10 + (a + 4)*31 + (a + 21)*31^2 + (11*a + 21)*31^3 + (21*a + 8)*31^4+O(31^5) $r_{ 5 }$ $=$ $$25 a + 10 + \left(29 a + 8\right)\cdot 31 + \left(29 a + 28\right)\cdot 31^{2} + \left(19 a + 10\right)\cdot 31^{3} + \left(9 a + 20\right)\cdot 31^{4} +O(31^{5})$$ 25*a + 10 + (29*a + 8)*31 + (29*a + 28)*31^2 + (19*a + 10)*31^3 + (9*a + 20)*31^4+O(31^5) $r_{ 6 }$ $=$ $$25 a + 18 + \left(29 a + 16\right)\cdot 31 + \left(29 a + 28\right)\cdot 31^{2} + \left(19 a + 8\right)\cdot 31^{3} + \left(9 a + 2\right)\cdot 31^{4} +O(31^{5})$$ 25*a + 18 + (29*a + 16)*31 + (29*a + 28)*31^2 + (19*a + 8)*31^3 + (9*a + 2)*31^4+O(31^5)

## Generators of the action on the roots $r_1, \ldots, r_{ 6 }$

 Cycle notation $(1,5)(2,4)(3,6)$ $(1,6,4,5,3,2)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $1$ $1$ $2$ $(1,5)(2,4)(3,6)$ $-1$ $1$ $3$ $(1,4,3)(2,6,5)$ $\zeta_{3}$ $1$ $3$ $(1,3,4)(2,5,6)$ $-\zeta_{3} - 1$ $1$ $6$ $(1,6,4,5,3,2)$ $\zeta_{3} + 1$ $1$ $6$ $(1,2,3,5,4,6)$ $-\zeta_{3}$

The blue line marks the conjugacy class containing complex conjugation.