# Properties

 Label 2.405.6t5.a.b Dimension $2$ Group $S_3\times C_3$ Conductor $405$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$405$$$$\medspace = 3^{4} \cdot 5$$ Artin stem field: 6.0.2460375.2 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.45.6t1.b.b Projective image: $S_3$ Projective stem field: 3.1.135.1

## Defining polynomial

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

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{2} + 33 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$14 a + 35 + \left(22 a + 13\right)\cdot 37 + \left(21 a + 1\right)\cdot 37^{2} + \left(32 a + 13\right)\cdot 37^{3} + \left(31 a + 10\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 2 }$ $=$ $$23 a + 17 + \left(14 a + 15\right)\cdot 37 + \left(15 a + 28\right)\cdot 37^{2} + \left(4 a + 10\right)\cdot 37^{3} + \left(5 a + 31\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 3 }$ $=$ $$18 a + 33 + 7\cdot 37 + \left(2 a + 25\right)\cdot 37^{2} + \left(17 a + 34\right)\cdot 37^{3} + \left(26 a + 4\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 4 }$ $=$ $$33 a + 24 + \left(21 a + 13\right)\cdot 37 + \left(19 a + 20\right)\cdot 37^{2} + \left(15 a + 28\right)\cdot 37^{3} + 5 a\cdot 37^{4} +O(37^{5})$$ $r_{ 5 }$ $=$ $$19 a + 31 + \left(36 a + 28\right)\cdot 37 + \left(34 a + 32\right)\cdot 37^{2} + \left(19 a + 26\right)\cdot 37^{3} + \left(10 a + 19\right)\cdot 37^{4} +O(37^{5})$$ $r_{ 6 }$ $=$ $$4 a + 8 + \left(15 a + 31\right)\cdot 37 + \left(17 a + 2\right)\cdot 37^{2} + \left(21 a + 34\right)\cdot 37^{3} + \left(31 a + 6\right)\cdot 37^{4} +O(37^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.