# Properties

 Label 1.344.6t1.d.a Dimension $1$ Group $C_6$ Conductor $344$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$344$$$$\medspace = 2^{3} \cdot 43$$ Artin number field: Galois closure of 6.6.75268322816.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{344}(179,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{6} - 86 x^{4} + 1376 x^{2} - 5504$.

The roots of $f$ are computed in an extension of $\Q_{ 47 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 47 }$: $x^{2} + 45 x + 5$

Roots:
 $r_{ 1 }$ $=$ $32 a + 15 + \left(a + 14\right)\cdot 47 + \left(22 a + 2\right)\cdot 47^{2} + \left(28 a + 6\right)\cdot 47^{3} + \left(29 a + 8\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 2 }$ $=$ $19 a + 28 + \left(24 a + 8\right)\cdot 47 + \left(28 a + 7\right)\cdot 47^{2} + \left(18 a + 19\right)\cdot 47^{3} + \left(40 a + 39\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 3 }$ $=$ $11 a + 36 + \left(46 a + 29\right)\cdot 47 + \left(29 a + 16\right)\cdot 47^{2} + \left(5 a + 9\right)\cdot 47^{3} + \left(a + 25\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 4 }$ $=$ $15 a + 32 + \left(45 a + 32\right)\cdot 47 + \left(24 a + 44\right)\cdot 47^{2} + \left(18 a + 40\right)\cdot 47^{3} + \left(17 a + 38\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 5 }$ $=$ $28 a + 19 + \left(22 a + 38\right)\cdot 47 + \left(18 a + 39\right)\cdot 47^{2} + \left(28 a + 27\right)\cdot 47^{3} + \left(6 a + 7\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$ $r_{ 6 }$ $=$ $36 a + 11 + 17\cdot 47 + \left(17 a + 30\right)\cdot 47^{2} + \left(41 a + 37\right)\cdot 47^{3} + \left(45 a + 21\right)\cdot 47^{4} +O\left(47^{ 5 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.