# Properties

 Label 1.45.6t1.b.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: 6.0.2460375.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{45}(14,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 6 x^{4} - 4 x^{3} + 9 x^{2} - 12 x + 19$$  .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$8 a + \left(23 a + 3\right)\cdot 37 + \left(28 a + 26\right)\cdot 37^{2} + \left(13 a + 35\right)\cdot 37^{3} + \left(20 a + 35\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 2 }$ $=$ $$29 a + 32 + \left(13 a + 13\right)\cdot 37 + \left(8 a + 6\right)\cdot 37^{2} + \left(23 a + 25\right)\cdot 37^{3} + \left(16 a + 29\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 3 }$ $=$ $$32 a + 24 + \left(30 a + 27\right)\cdot 37 + \left(18 a + 17\right)\cdot 37^{2} + \left(10 a + 16\right)\cdot 37^{3} + \left(23 a + 34\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 4 }$ $=$ $$3 a + 1 + \left(17 a + 15\right)\cdot 37 + \left(10 a + 5\right)\cdot 37^{2} + \left(24 a + 9\right)\cdot 37^{3} + \left(6 a + 1\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 5 }$ $=$ $$5 a + 4 + \left(6 a + 8\right)\cdot 37 + \left(18 a + 25\right)\cdot 37^{2} + \left(26 a + 2\right)\cdot 37^{3} + \left(13 a + 6\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 6 }$ $=$ $$34 a + 13 + \left(19 a + 6\right)\cdot 37 + \left(26 a + 30\right)\cdot 37^{2} + \left(12 a + 21\right)\cdot 37^{3} + \left(30 a + 3\right)\cdot 37^{4} +O(37^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.