# Properties

 Label 12.223...504.18t206.a.a Dimension $12$ Group $S_3 \wr C_3$ Conductor $2.236\times 10^{17}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $12$ Group: $S_3 \wr C_3$ Conductor: $$223580268118933504$$$$\medspace = 2^{18} \cdot 31^{8}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.3.28400117792.1 Galois orbit size: $1$ Smallest permutation container: 18T206 Parity: even Determinant: 1.1.1t1.a.a Projective image: $C_3^3:C_2^2.C_6$ Projective stem field: Galois closure of 9.3.28400117792.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 3x^{8} + 5x^{7} - 8x^{6} - x^{5} + 2x^{4} + 5x^{3} + 6x^{2} + 2x - 1$$ x^9 - 3*x^8 + 5*x^7 - 8*x^6 - x^5 + 2*x^4 + 5*x^3 + 6*x^2 + 2*x - 1 .

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 11 }$: $$x^{3} + 2x + 9$$

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

## Generators of the action on the roots $r_1, \ldots, r_{ 9 }$

 Cycle notation $(8,9)$ $(1,2,4)(3,8,6)(5,9,7)$ $(2,3,5)$ $(6,7)$ $(4,8,9)$ $(1,6,7)$ $(2,3)$

## Character values on conjugacy classes

 Size Order Action on $r_1, \ldots, r_{ 9 }$ Character value $1$ $1$ $()$ $12$ $9$ $2$ $(2,3)$ $4$ $27$ $2$ $(1,6)(2,3)(4,8)$ $0$ $27$ $2$ $(2,3)(6,7)$ $0$ $6$ $3$ $(4,8,9)$ $0$ $8$ $3$ $(1,6,7)(2,3,5)(4,8,9)$ $3$ $12$ $3$ $(1,6,7)(4,8,9)$ $-3$ $36$ $3$ $(1,2,4)(3,8,6)(5,9,7)$ $0$ $36$ $3$ $(1,4,2)(3,6,8)(5,7,9)$ $0$ $18$ $6$ $(2,3)(4,8,9)$ $-2$ $18$ $6$ $(1,6,7)(2,3)$ $-2$ $36$ $6$ $(1,6,7)(2,3)(4,8,9)$ $1$ $54$ $6$ $(2,3)(4,8,9)(6,7)$ $0$ $108$ $6$ $(1,2,8,6,3,4)(5,9,7)$ $0$ $108$ $6$ $(1,4,3,6,8,2)(5,7,9)$ $0$ $72$ $9$ $(1,2,4,6,3,8,7,5,9)$ $0$ $72$ $9$ $(1,4,3,7,9,2,6,8,5)$ $0$

The blue line marks the conjugacy class containing complex conjugation.