# Properties

 Label 2.2888.6t5.a.b Dimension $2$ Group $S_3\times C_3$ Conductor $2888$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: $S_3\times C_3$ Conductor: $$2888$$$$\medspace = 2^{3} \cdot 19^{2}$$ Artin stem field: Galois closure of 6.0.1267762688.3 Galois orbit size: $2$ Smallest permutation container: $S_3\times C_3$ Parity: odd Determinant: 1.152.6t1.d.a Projective image: $S_3$ Projective stem field: Galois closure of 3.1.152.1

## Defining polynomial

 $f(x)$ $=$ $$x^{6} - 2x^{5} - 11x^{4} + 26x^{3} + 174x^{2} - 388x + 201$$ x^6 - 2*x^5 - 11*x^4 + 26*x^3 + 174*x^2 - 388*x + 201 .

The roots of $f$ are computed in an extension of $\Q_{ 11 }$ to precision 9.

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

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

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.