# Properties

 Label 2.2268.6t5.h.b Dimension $2$ Group $S_3\times C_3$ Conductor $2268$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$2268$$$$\medspace = 2^{2} \cdot 3^{4} \cdot 7$$ Artin stem field: 6.0.15431472.1 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.63.6t1.a.b Projective image: $S_3$ Projective stem field: 3.1.588.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 10 x^{3} + 28$$  .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{2} + 7 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$3 a + 3 + \left(7 a + 8\right)\cdot 11 + \left(8 a + 9\right)\cdot 11^{2} + 11^{3} + \left(9 a + 7\right)\cdot 11^{4} + \left(10 a + 2\right)\cdot 11^{5} + \left(10 a + 9\right)\cdot 11^{6} + \left(6 a + 4\right)\cdot 11^{7} + \left(8 a + 5\right)\cdot 11^{8} + \left(3 a + 1\right)\cdot 11^{9} +O(11^{10})$$ $r_{ 2 }$ $=$ $$8 a + 4 + \left(3 a + 1\right)\cdot 11 + \left(2 a + 4\right)\cdot 11^{2} + \left(10 a + 7\right)\cdot 11^{3} + \left(a + 9\right)\cdot 11^{4} + 3\cdot 11^{5} + 9\cdot 11^{6} + \left(4 a + 10\right)\cdot 11^{7} + \left(2 a + 10\right)\cdot 11^{8} + \left(7 a + 7\right)\cdot 11^{9} +O(11^{10})$$ $r_{ 3 }$ $=$ $$6 a + 5 + 7 a\cdot 11 + \left(10 a + 6\right)\cdot 11^{2} + \left(7 a + 6\right)\cdot 11^{3} + \left(8 a + 7\right)\cdot 11^{4} + \left(4 a + 10\right)\cdot 11^{5} + \left(5 a + 7\right)\cdot 11^{6} + \left(10 a + 5\right)\cdot 11^{7} + \left(8 a + 8\right)\cdot 11^{8} + 4\cdot 11^{9} +O(11^{10})$$ $r_{ 4 }$ $=$ $$9 a + \left(3 a + 7\right)\cdot 11 + \left(8 a + 9\right)\cdot 11^{2} + \left(8 a + 8\right)\cdot 11^{3} + \left(6 a + 10\right)\cdot 11^{4} + \left(4 a + 7\right)\cdot 11^{5} + \left(5 a + 9\right)\cdot 11^{6} + \left(6 a + 1\right)\cdot 11^{7} + \left(6 a + 10\right)\cdot 11^{8} + \left(4 a + 3\right)\cdot 11^{9} +O(11^{10})$$ $r_{ 5 }$ $=$ $$2 a + 3 + \left(7 a + 2\right)\cdot 11 + \left(2 a + 6\right)\cdot 11^{2} + \left(2 a + 2\right)\cdot 11^{3} + \left(4 a + 7\right)\cdot 11^{4} + \left(6 a + 8\right)\cdot 11^{5} + \left(5 a + 4\right)\cdot 11^{6} + 4 a\cdot 11^{7} + \left(4 a + 8\right)\cdot 11^{8} + \left(6 a + 4\right)\cdot 11^{9} +O(11^{10})$$ $r_{ 6 }$ $=$ $$5 a + 7 + \left(3 a + 2\right)\cdot 11 + 8\cdot 11^{2} + \left(3 a + 5\right)\cdot 11^{3} + \left(2 a + 1\right)\cdot 11^{4} + \left(6 a + 10\right)\cdot 11^{5} + \left(5 a + 2\right)\cdot 11^{6} + 9\cdot 11^{7} + 2 a\cdot 11^{8} + \left(10 a + 10\right)\cdot 11^{9} +O(11^{10})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.