# Properties

 Label 2.280.6t5.a.b Dimension $2$ Group $S_3\times C_3$ Conductor $280$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$280$$$$\medspace = 2^{3} \cdot 5 \cdot 7$$ Artin stem field: 6.0.3136000.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.280.6t1.d.b Projective image: $S_3$ Projective stem field: 3.1.1960.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2 x^{5} + x^{4} - 2 x^{3} + 12 x^{2} - 20 x + 11$$  .

The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 5.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$: $$x^{2} + 24 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$20 a + 20 + \left(2 a + 7\right)\cdot 29 + \left(3 a + 10\right)\cdot 29^{2} + \left(3 a + 22\right)\cdot 29^{3} + \left(7 a + 11\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 2 }$ $=$ $$14 a + 8 + \left(4 a + 6\right)\cdot 29 + \left(16 a + 28\right)\cdot 29^{2} + \left(10 a + 14\right)\cdot 29^{3} + \left(a + 9\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 3 }$ $=$ $$24 a + 2 + \left(21 a + 15\right)\cdot 29 + \left(9 a + 19\right)\cdot 29^{2} + \left(15 a + 20\right)\cdot 29^{3} + \left(20 a + 7\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 4 }$ $=$ $$5 a + 6 + \left(7 a + 13\right)\cdot 29 + \left(19 a + 17\right)\cdot 29^{2} + 13 a\cdot 29^{3} + \left(8 a + 8\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 5 }$ $=$ $$9 a + 4 + \left(26 a + 1\right)\cdot 29 + \left(25 a + 23\right)\cdot 29^{2} + \left(25 a + 5\right)\cdot 29^{3} + \left(21 a + 15\right)\cdot 29^{4} +O(29^{5})$$ $r_{ 6 }$ $=$ $$15 a + 20 + \left(24 a + 14\right)\cdot 29 + \left(12 a + 17\right)\cdot 29^{2} + \left(18 a + 22\right)\cdot 29^{3} + \left(27 a + 5\right)\cdot 29^{4} +O(29^{5})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.