# Properties

 Label 1.7.6t1.a.b Dimension $1$ Group $C_6$ Conductor $7$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$7$$ Artin field: $$\Q(\zeta_{7})$$ Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{7}(5,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{2} + 12 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$6 a + 6 + \left(9 a + 3\right)\cdot 13 + \left(2 a + 5\right)\cdot 13^{2} + \left(3 a + 10\right)\cdot 13^{3} + \left(10 a + 2\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 2 }$ $=$ $$a + 9 + \left(8 a + 9\right)\cdot 13 + 10 a\cdot 13^{2} + \left(a + 4\right)\cdot 13^{3} + \left(2 a + 7\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 3 }$ $=$ $$4 a + 6 + \left(3 a + 1\right)\cdot 13 + \left(9 a + 6\right)\cdot 13^{2} + \left(2 a + 12\right)\cdot 13^{3} + \left(5 a + 10\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 4 }$ $=$ $$9 a + 10 + 9 a\cdot 13 + \left(3 a + 12\right)\cdot 13^{2} + \left(10 a + 5\right)\cdot 13^{3} + 7 a\cdot 13^{4} +O(13^{5})$$ $r_{ 5 }$ $=$ $$12 a + 10 + \left(4 a + 3\right)\cdot 13 + \left(2 a + 3\right)\cdot 13^{2} + \left(11 a + 8\right)\cdot 13^{3} + \left(10 a + 7\right)\cdot 13^{4} +O(13^{5})$$ $r_{ 6 }$ $=$ $$7 a + 12 + \left(3 a + 6\right)\cdot 13 + \left(10 a + 11\right)\cdot 13^{2} + \left(9 a + 10\right)\cdot 13^{3} + \left(2 a + 9\right)\cdot 13^{4} +O(13^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.