# Properties

 Label 1.45.6t1.a.b Dimension $1$ Group $C_6$ Conductor $45$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$45$$$$\medspace = 3^{2} \cdot 5$$ Artin field: 6.6.820125.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{45}(4,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$$  .

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:
 $r_{ 1 }$ $=$ $$11 a + 5 + \left(3 a + 8\right)\cdot 17 + \left(a + 16\right)\cdot 17^{2} + \left(14 a + 15\right)\cdot 17^{3} + \left(14 a + 15\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$13 a + 12 + \left(15 a + 4\right)\cdot 17 + \left(7 a + 11\right)\cdot 17^{2} + \left(11 a + 5\right)\cdot 17^{3} + \left(12 a + 10\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 3 }$ $=$ $$4 a + 8 + \left(a + 7\right)\cdot 17 + \left(9 a + 3\right)\cdot 17^{2} + \left(5 a + 9\right)\cdot 17^{3} + \left(4 a + 11\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$15 a + 6 + \left(4 a + 11\right)\cdot 17 + \left(10 a + 8\right)\cdot 17^{2} + \left(2 a + 16\right)\cdot 17^{3} + \left(2 a + 6\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 5 }$ $=$ $$2 a + 4 + \left(12 a + 1\right)\cdot 17 + \left(6 a + 14\right)\cdot 17^{2} + \left(14 a + 8\right)\cdot 17^{3} + \left(14 a + 6\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 6 }$ $=$ $$6 a + 16 + 13 a\cdot 17 + \left(15 a + 14\right)\cdot 17^{2} + \left(2 a + 11\right)\cdot 17^{3} + \left(2 a + 16\right)\cdot 17^{4} +O(17^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.