# Properties

 Label 2.1444.6t5.b.b Dimension $2$ Group $S_3\times C_3$ Conductor $1444$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$1444$$$$\medspace = 2^{2} \cdot 19^{2}$$ Artin stem field: 6.0.39617584.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.19.6t1.a.a Projective image: $S_3$ Projective stem field: 3.1.76.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - x^{5} + 2 x^{4} - 11 x^{3} + 18 x^{2} + 33 x + 83$$  .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 6.

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

Roots:
 $r_{ 1 }$ $=$ $$28 a + 3 + \left(2 a + 8\right)\cdot 31 + \left(10 a + 20\right)\cdot 31^{2} + \left(29 a + 13\right)\cdot 31^{3} + 3 a\cdot 31^{4} + \left(12 a + 14\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 2 }$ $=$ $$4 a + 6 + \left(2 a + 18\right)\cdot 31 + \left(14 a + 2\right)\cdot 31^{2} + \left(5 a + 21\right)\cdot 31^{3} + \left(26 a + 2\right)\cdot 31^{4} + \left(a + 12\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 3 }$ $=$ $$3 a + 28 + \left(28 a + 16\right)\cdot 31 + \left(20 a + 6\right)\cdot 31^{2} + a\cdot 31^{3} + \left(27 a + 10\right)\cdot 31^{4} + \left(18 a + 3\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 4 }$ $=$ $$6 a + \left(12 a + 22\right)\cdot 31 + \left(16 a + 22\right)\cdot 31^{2} + \left(5 a + 22\right)\cdot 31^{3} + \left(4 a + 13\right)\cdot 31^{4} + \left(18 a + 5\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 5 }$ $=$ $$27 a + 14 + \left(28 a + 18\right)\cdot 31 + \left(16 a + 28\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(4 a + 18\right)\cdot 31^{4} + \left(29 a + 20\right)\cdot 31^{5} +O(31^{6})$$ $r_{ 6 }$ $=$ $$25 a + 12 + \left(18 a + 9\right)\cdot 31 + \left(14 a + 12\right)\cdot 31^{2} + \left(25 a + 17\right)\cdot 31^{3} + \left(26 a + 16\right)\cdot 31^{4} + \left(12 a + 6\right)\cdot 31^{5} +O(31^{6})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.