# Properties

 Label 2.648.6t5.b.b Dimension $2$ Group $S_3\times C_3$ Conductor $648$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$648$$$$\medspace = 2^{3} \cdot 3^{4}$$ Artin stem field: 6.0.10077696.3 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.72.6t1.c.b Projective image: $S_3$ Projective stem field: 3.1.216.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} + 6 x^{4} - 2 x^{3} + 9 x^{2} - 6 x + 7$$  .

The roots of $f$ are computed in an extension of $\Q_{ 17 }$ to precision 6.

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

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

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

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

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 6 }$ Character value $1$ $1$ $()$ $2$ $3$ $2$ $(1,5)(2,6)(3,4)$ $0$ $1$ $3$ $(1,6,3)(2,4,5)$ $-2 \zeta_{3} - 2$ $1$ $3$ $(1,3,6)(2,5,4)$ $2 \zeta_{3}$ $2$ $3$ $(1,6,3)$ $-\zeta_{3}$ $2$ $3$ $(1,3,6)$ $\zeta_{3} + 1$ $2$ $3$ $(1,3,6)(2,4,5)$ $-1$ $3$ $6$ $(1,2,6,4,3,5)$ $0$ $3$ $6$ $(1,5,3,4,6,2)$ $0$

The blue line marks the conjugacy class containing complex conjugation.