# Properties

 Label 1.456.6t1.b.a Dimension $1$ Group $C_6$ Conductor $456$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$456$$$$\medspace = 2^{3} \cdot 3 \cdot 19$$ Artin field: Galois closure of 6.6.1801557504.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{456}(11,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} - 29x^{4} + 50x^{3} + 118x^{2} + 24x + 1$$ x^6 - 2*x^5 - 29*x^4 + 50*x^3 + 118*x^2 + 24*x + 1 .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$a + 1 + 2\cdot 7 + 6\cdot 7^{2} + 5\cdot 7^{3} + 2\cdot 7^{4} +O(7^{5})$$ a + 1 + 2*7 + 6*7^2 + 5*7^3 + 2*7^4+O(7^5) $r_{ 2 }$ $=$ $$6 a + 4 + \left(6 a + 5\right)\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + 6 a\cdot 7^{3} + \left(6 a + 6\right)\cdot 7^{4} +O(7^{5})$$ 6*a + 4 + (6*a + 5)*7 + (6*a + 6)*7^2 + 6*a*7^3 + (6*a + 6)*7^4+O(7^5) $r_{ 3 }$ $=$ $$6 a + \left(6 a + 6\right)\cdot 7 + 6 a\cdot 7^{2} + 6 a\cdot 7^{3} + \left(6 a + 5\right)\cdot 7^{4} +O(7^{5})$$ 6*a + (6*a + 6)*7 + 6*a*7^2 + 6*a*7^3 + (6*a + 5)*7^4+O(7^5) $r_{ 4 }$ $=$ $$a + 6 + 6\cdot 7 + 5\cdot 7^{4} +O(7^{5})$$ a + 6 + 6*7 + 5*7^4+O(7^5) $r_{ 5 }$ $=$ $$6 a + 2 + \left(6 a + 1\right)\cdot 7 + \left(6 a + 6\right)\cdot 7^{2} + \left(6 a + 5\right)\cdot 7^{3} + \left(6 a + 2\right)\cdot 7^{4} +O(7^{5})$$ 6*a + 2 + (6*a + 1)*7 + (6*a + 6)*7^2 + (6*a + 5)*7^3 + (6*a + 2)*7^4+O(7^5) $r_{ 6 }$ $=$ $$a + 3 + 6\cdot 7 + 6\cdot 7^{2} + 6\cdot 7^{4} +O(7^{5})$$ a + 3 + 6*7 + 6*7^2 + 6*7^4+O(7^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.