# Properties

 Label 1.760.6t1.a.b Dimension $1$ Group $C_6$ Conductor $760$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$760$$$$\medspace = 2^{3} \cdot 5 \cdot 19$$ Artin field: 6.6.8340544000.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{760}(349,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2 x^{5} - 41 x^{4} + 66 x^{3} + 302 x^{2} + 16 x - 151$$  .

The roots of $f$ are computed in an extension of $\Q_{ 7 }$ to precision 6.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 7 }$: $$x^{2} + 6 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$5 a + 4 + 2 a\cdot 7 + \left(3 a + 4\right)\cdot 7^{2} + 2 a\cdot 7^{3} + 5 a\cdot 7^{4} + \left(3 a + 6\right)\cdot 7^{5} +O(7^{6})$$ $r_{ 2 }$ $=$ $$2 a + 6 + \left(4 a + 4\right)\cdot 7 + \left(3 a + 3\right)\cdot 7^{2} + 4 a\cdot 7^{3} + \left(a + 4\right)\cdot 7^{4} + 3 a\cdot 7^{5} +O(7^{6})$$ $r_{ 3 }$ $=$ $$2 a + 4 + 4 a\cdot 7 + \left(3 a + 3\right)\cdot 7^{2} + \left(4 a + 5\right)\cdot 7^{3} + a\cdot 7^{4} + \left(3 a + 3\right)\cdot 7^{5} +O(7^{6})$$ $r_{ 4 }$ $=$ $$2 a + 2 + \left(4 a + 5\right)\cdot 7 + \left(3 a + 4\right)\cdot 7^{2} + \left(4 a + 6\right)\cdot 7^{3} + \left(a + 2\right)\cdot 7^{4} + \left(3 a + 4\right)\cdot 7^{5} +O(7^{6})$$ $r_{ 5 }$ $=$ $$5 a + 1 + 2 a\cdot 7 + \left(3 a + 3\right)\cdot 7^{2} + \left(2 a + 1\right)\cdot 7^{3} + \left(5 a + 1\right)\cdot 7^{4} + \left(3 a + 2\right)\cdot 7^{5} +O(7^{6})$$ $r_{ 6 }$ $=$ $$5 a + 6 + \left(2 a + 2\right)\cdot 7 + \left(3 a + 2\right)\cdot 7^{2} + \left(2 a + 6\right)\cdot 7^{3} + \left(5 a + 4\right)\cdot 7^{4} + \left(3 a + 4\right)\cdot 7^{5} +O(7^{6})$$

## 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.