# Properties

 Label 1.91.6t1.j.a Dimension $1$ Group $C_6$ Conductor $91$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$91$$$$\medspace = 7 \cdot 13$$ Artin field: 6.6.5274997.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: even Dirichlet character: $$\chi_{91}(25,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} - 14 x^{4} + 9 x^{3} + 35 x^{2} - 16 x - 1$$  .

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:
 $r_{ 1 }$ $=$ $$12 a + 40 + \left(29 a + 29\right)\cdot 41 + \left(40 a + 29\right)\cdot 41^{2} + 9\cdot 41^{3} + \left(37 a + 37\right)\cdot 41^{4} +O(41^{5})$$ $r_{ 2 }$ $=$ $$12 a + 17 + \left(29 a + 33\right)\cdot 41 + \left(40 a + 11\right)\cdot 41^{2} + 25\cdot 41^{3} + \left(37 a + 15\right)\cdot 41^{4} +O(41^{5})$$ $r_{ 3 }$ $=$ $$29 a + 28 + \left(11 a + 21\right)\cdot 41 + 14\cdot 41^{2} + \left(40 a + 4\right)\cdot 41^{3} + \left(3 a + 15\right)\cdot 41^{4} +O(41^{5})$$ $r_{ 4 }$ $=$ $$29 a + 12 + \left(11 a + 27\right)\cdot 41 + 22\cdot 41^{2} + \left(40 a + 28\right)\cdot 41^{3} + \left(3 a + 2\right)\cdot 41^{4} +O(41^{5})$$ $r_{ 5 }$ $=$ $$29 a + 35 + \left(11 a + 23\right)\cdot 41 + 40\cdot 41^{2} + \left(40 a + 12\right)\cdot 41^{3} + \left(3 a + 24\right)\cdot 41^{4} +O(41^{5})$$ $r_{ 6 }$ $=$ $$12 a + 33 + \left(29 a + 27\right)\cdot 41 + \left(40 a + 3\right)\cdot 41^{2} + 41^{3} + \left(37 a + 28\right)\cdot 41^{4} +O(41^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.