# Properties

 Label 1.99.6t1.a.a Dimension $1$ Group $C_6$ Conductor $99$ Root number not computed Indicator $0$

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## Basic invariants

 Dimension: $1$ Group: $C_6$ Conductor: $$99$$$$\medspace = 3^{2} \cdot 11$$ Artin field: 6.0.8732691.1 Galois orbit size: $2$ Smallest permutation container: $C_6$ Parity: odd Dirichlet character: $$\chi_{99}(76,\cdot)$$ Projective image: $C_1$ Projective field: $$\Q$$

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 17 }$: $$x^{2} + 16 x + 3$$

Roots:
 $r_{ 1 }$ $=$ $$a + 14 + \left(3 a + 3\right)\cdot 17 + \left(a + 3\right)\cdot 17^{2} + \left(16 a + 3\right)\cdot 17^{3} + \left(3 a + 1\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 2 }$ $=$ $$16 a + 14 + \left(13 a + 8\right)\cdot 17 + \left(15 a + 2\right)\cdot 17^{2} + 5\cdot 17^{3} + \left(13 a + 12\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 3 }$ $=$ $$16 a + 15 + \left(13 a + 5\right)\cdot 17 + \left(15 a + 1\right)\cdot 17^{2} + 17^{3} + \left(13 a + 6\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 4 }$ $=$ $$a + 13 + \left(3 a + 6\right)\cdot 17 + \left(a + 4\right)\cdot 17^{2} + \left(16 a + 7\right)\cdot 17^{3} + \left(3 a + 7\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 5 }$ $=$ $$16 a + 8 + \left(13 a + 5\right)\cdot 17 + \left(15 a + 10\right)\cdot 17^{2} + 7\cdot 17^{3} + \left(13 a + 14\right)\cdot 17^{4} +O(17^{5})$$ $r_{ 6 }$ $=$ $$a + 7 + \left(3 a + 3\right)\cdot 17 + \left(a + 12\right)\cdot 17^{2} + \left(16 a + 9\right)\cdot 17^{3} + \left(3 a + 9\right)\cdot 17^{4} +O(17^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.