# Properties

 Label 1.57.6t1.a.a Dimension $1$ Group $C_6$ Conductor $57$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$57$$$$\medspace = 3 \cdot 19$$ Artin field: Galois closure of 6.0.3518667.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{57}(26,\cdot)$$ Projective image: $C_1$ Projective field: Galois closure of $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 7x^{4} - 8x^{3} + 43x^{2} - 42x + 49$$ x^6 - x^5 + 7*x^4 - 8*x^3 + 43*x^2 - 42*x + 49 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$8 a + 10 + \left(6 a + 5\right)\cdot 11 + \left(a + 4\right)\cdot 11^{2} + \left(7 a + 10\right)\cdot 11^{3} + \left(10 a + 7\right)\cdot 11^{4} +O(11^{5})$$ 8*a + 10 + (6*a + 5)*11 + (a + 4)*11^2 + (7*a + 10)*11^3 + (10*a + 7)*11^4+O(11^5) $r_{ 2 }$ $=$ $$5 a + 9 + \left(8 a + 2\right)\cdot 11 + \left(6 a + 10\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(7 a + 5\right)\cdot 11^{4} +O(11^{5})$$ 5*a + 9 + (8*a + 2)*11 + (6*a + 10)*11^2 + (10*a + 7)*11^3 + (7*a + 5)*11^4+O(11^5) $r_{ 3 }$ $=$ $$10 a + 7 + \left(3 a + 8\right)\cdot 11 + \left(3 a + 9\right)\cdot 11^{2} + \left(8 a + 6\right)\cdot 11^{3} + \left(9 a + 8\right)\cdot 11^{4} +O(11^{5})$$ 10*a + 7 + (3*a + 8)*11 + (3*a + 9)*11^2 + (8*a + 6)*11^3 + (9*a + 8)*11^4+O(11^5) $r_{ 4 }$ $=$ $$3 a + 9 + \left(4 a + 2\right)\cdot 11 + \left(9 a + 4\right)\cdot 11^{2} + \left(3 a + 4\right)\cdot 11^{3} + 10\cdot 11^{4} +O(11^{5})$$ 3*a + 9 + (4*a + 2)*11 + (9*a + 4)*11^2 + (3*a + 4)*11^3 + 10*11^4+O(11^5) $r_{ 5 }$ $=$ $$6 a + 7 + \left(2 a + 9\right)\cdot 11 + \left(4 a + 6\right)\cdot 11^{2} + 10\cdot 11^{3} + \left(3 a + 4\right)\cdot 11^{4} +O(11^{5})$$ 6*a + 7 + (2*a + 9)*11 + (4*a + 6)*11^2 + 10*11^3 + (3*a + 4)*11^4+O(11^5) $r_{ 6 }$ $=$ $$a + 3 + \left(7 a + 3\right)\cdot 11 + \left(7 a + 8\right)\cdot 11^{2} + \left(2 a + 3\right)\cdot 11^{3} + \left(a + 6\right)\cdot 11^{4} +O(11^{5})$$ a + 3 + (7*a + 3)*11 + (7*a + 8)*11^2 + (2*a + 3)*11^3 + (a + 6)*11^4+O(11^5)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.