# Properties

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

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## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$36$$$$\medspace = 2^{2} \cdot 3^{2}$$ Artin field: 6.0.419904.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{36}(31,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 6 x^{4} + 9 x^{2} + 1$$  .

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $$x^{2} + 18 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$16 a + 11 + \left(10 a + 2\right)\cdot 19 + \left(3 a + 13\right)\cdot 19^{2} + \left(3 a + 9\right)\cdot 19^{3} + \left(11 a + 5\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 2 }$ $=$ $$7 a + 6 + \left(7 a + 9\right)\cdot 19 + \left(12 a + 16\right)\cdot 19^{2} + \left(16 a + 16\right)\cdot 19^{3} + \left(3 a + 15\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 3 }$ $=$ $$15 a + 2 + 7\cdot 19 + \left(3 a + 8\right)\cdot 19^{2} + \left(18 a + 11\right)\cdot 19^{3} + \left(3 a + 16\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 4 }$ $=$ $$3 a + 8 + \left(8 a + 16\right)\cdot 19 + \left(15 a + 5\right)\cdot 19^{2} + \left(15 a + 9\right)\cdot 19^{3} + \left(7 a + 13\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 5 }$ $=$ $$12 a + 13 + \left(11 a + 9\right)\cdot 19 + \left(6 a + 2\right)\cdot 19^{2} + \left(2 a + 2\right)\cdot 19^{3} + \left(15 a + 3\right)\cdot 19^{4} +O(19^{5})$$ $r_{ 6 }$ $=$ $$4 a + 17 + \left(18 a + 11\right)\cdot 19 + \left(15 a + 10\right)\cdot 19^{2} + 7\cdot 19^{3} + \left(15 a + 2\right)\cdot 19^{4} +O(19^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.