# Properties

 Label 1.36.6t1.a.b Dimension $1$ Group $C_6$ Conductor $36$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$36$$$$\medspace = 2^{2} \cdot 3^{2}$$ Artin field: $$\Q(\zeta_{36})^+$$ Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{36}(11,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 6 x^{4} + 9 x^{2} - 3$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$14 a + 10 + \left(5 a + 12\right)\cdot 17 + \left(11 a + 5\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(12 a + 11\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$12 a + 11 + \left(3 a + 12\right)\cdot 17 + \left(9 a + 5\right)\cdot 17^{2} + \left(2 a + 3\right)\cdot 17^{3} + \left(6 a + 15\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 3 }$ $=$ $$2 a + 16 + \left(2 a + 16\right)\cdot 17 + \left(2 a + 16\right)\cdot 17^{2} + \left(15 a + 1\right)\cdot 17^{3} + \left(5 a + 13\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$3 a + 7 + \left(11 a + 4\right)\cdot 17 + \left(5 a + 11\right)\cdot 17^{2} + \left(16 a + 11\right)\cdot 17^{3} + \left(4 a + 5\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 5 }$ $=$ $$5 a + 6 + \left(13 a + 4\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(14 a + 13\right)\cdot 17^{3} + \left(10 a + 1\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 6 }$ $=$ $$15 a + 1 + 14 a\cdot 17 + 14 a\cdot 17^{2} + \left(a + 15\right)\cdot 17^{3} + \left(11 a + 3\right)\cdot 17^{4} +O(17^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.