# Properties

 Label 1.455.6t1.b.a Dimension $1$ Group $C_6$ Conductor $455$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$455$$$$\medspace = 5 \cdot 7 \cdot 13$$ Artin field: 6.0.1224552875.2 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{455}(419,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 18 x^{4} - 7 x^{3} + 249 x^{2} - 230 x + 1535$$  .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $$x^{2} + 29 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$4 a + 21 + \left(13 a + 15\right)\cdot 31 + \left(24 a + 19\right)\cdot 31^{2} + 20 a\cdot 31^{3} + \left(25 a + 18\right)\cdot 31^{4} +O(31^{5})$$ $r_{ 2 }$ $=$ $$27 a + 16 + \left(17 a + 20\right)\cdot 31 + \left(6 a + 17\right)\cdot 31^{2} + \left(10 a + 29\right)\cdot 31^{3} + \left(5 a + 9\right)\cdot 31^{4} +O(31^{5})$$ $r_{ 3 }$ $=$ $$27 a + 29 + \left(17 a + 6\right)\cdot 31 + \left(6 a + 24\right)\cdot 31^{2} + \left(10 a + 17\right)\cdot 31^{3} + \left(5 a + 17\right)\cdot 31^{4} +O(31^{5})$$ $r_{ 4 }$ $=$ $$4 a + 6 + \left(13 a + 30\right)\cdot 31 + \left(24 a + 6\right)\cdot 31^{2} + \left(20 a + 23\right)\cdot 31^{3} + \left(25 a + 18\right)\cdot 31^{4} +O(31^{5})$$ $r_{ 5 }$ $=$ $$4 a + 8 + \left(13 a + 29\right)\cdot 31 + \left(24 a + 12\right)\cdot 31^{2} + \left(20 a + 12\right)\cdot 31^{3} + \left(25 a + 10\right)\cdot 31^{4} +O(31^{5})$$ $r_{ 6 }$ $=$ $$27 a + 14 + \left(17 a + 21\right)\cdot 31 + \left(6 a + 11\right)\cdot 31^{2} + \left(10 a + 9\right)\cdot 31^{3} + \left(5 a + 18\right)\cdot 31^{4} +O(31^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.