# Properties

 Label 1.35.6t1.a.b Dimension $1$ Group $C_6$ Conductor $35$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$35$$$$\medspace = 5 \cdot 7$$ Artin number field: Galois closure of 6.0.2100875.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{35}(24,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $x^{6} - x^{5} + 8 x^{4} - 8 x^{3} + 22 x^{2} - 22 x + 29$.

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $x^{2} + 38 x + 6$

Roots:
 $r_{ 1 }$ $=$ $10 a + 19 + \left(33 a + 1\right)\cdot 41 + \left(27 a + 38\right)\cdot 41^{2} + \left(14 a + 39\right)\cdot 41^{3} + \left(9 a + 39\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 2 }$ $=$ $23 a + 29 + \left(40 a + 39\right)\cdot 41 + \left(22 a + 39\right)\cdot 41^{2} + \left(15 a + 2\right)\cdot 41^{3} + \left(8 a + 31\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 3 }$ $=$ $18 a + 16 + 15\cdot 41 + \left(18 a + 27\right)\cdot 41^{2} + \left(25 a + 26\right)\cdot 41^{3} + \left(32 a + 40\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 4 }$ $=$ $17 a + 21 + \left(27 a + 16\right)\cdot 41 + \left(34 a + 8\right)\cdot 41^{2} + \left(28 a + 34\right)\cdot 41^{3} + \left(7 a + 22\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 5 }$ $=$ $24 a + 31 + \left(13 a + 40\right)\cdot 41 + \left(6 a + 2\right)\cdot 41^{2} + \left(12 a + 4\right)\cdot 41^{3} + \left(33 a + 17\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$ $r_{ 6 }$ $=$ $31 a + 8 + \left(7 a + 9\right)\cdot 41 + \left(13 a + 6\right)\cdot 41^{2} + \left(26 a + 15\right)\cdot 41^{3} + \left(31 a + 12\right)\cdot 41^{4} +O\left(41^{ 5 }\right)$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.