# Properties

 Label 12.118...757.18t218.a.a Dimension $12$ Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor $1.181\times 10^{17}$ Root number $1$ Indicator $1$

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## Basic invariants

 Dimension: $12$ Group: $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ Conductor: $$118129876015852757$$$$\medspace = 7^{10} \cdot 53^{5}$$ Frobenius-Schur indicator: $1$ Root number: $1$ Artin stem field: Galois closure of 9.5.45487052759281.1 Galois orbit size: $1$ Smallest permutation container: 18T218 Parity: even Determinant: 1.53.2t1.a.a Projective image: $C_3^3:S_4$ Projective stem field: Galois closure of 9.5.45487052759281.1

## Defining polynomial

 $f(x)$ $=$ $$x^{9} - 7x^{7} - 7x^{6} - 7x^{5} + 84x^{4} + 35x^{3} - 280x^{2} + 28x + 280$$ x^9 - 7*x^7 - 7*x^6 - 7*x^5 + 84*x^4 + 35*x^3 - 280*x^2 + 28*x + 280 .

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: $$x^{3} + 2x + 11$$

Roots:
 $r_{ 1 }$ $=$ $$4 a^{2} + 9 a + 10 + \left(11 a^{2} + 9 a + 7\right)\cdot 13 + \left(3 a^{2} + 11 a + 8\right)\cdot 13^{2} + \left(a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(12 a^{2} + 8 a\right)\cdot 13^{5} + \left(a^{2} + 9 a + 9\right)\cdot 13^{6} + \left(12 a + 6\right)\cdot 13^{7} + \left(a^{2} + 9 a + 4\right)\cdot 13^{8} + \left(11 a^{2} + 4 a + 10\right)\cdot 13^{9} +O(13^{10})$$ 4*a^2 + 9*a + 10 + (11*a^2 + 9*a + 7)*13 + (3*a^2 + 11*a + 8)*13^2 + (a^2 + 5*a + 12)*13^3 + (a^2 + 4*a + 11)*13^4 + (12*a^2 + 8*a)*13^5 + (a^2 + 9*a + 9)*13^6 + (12*a + 6)*13^7 + (a^2 + 9*a + 4)*13^8 + (11*a^2 + 4*a + 10)*13^9+O(13^10) $r_{ 2 }$ $=$ $$10 a^{2} + 11 a + 10 + \left(11 a^{2} + 11 a + 10\right)\cdot 13 + \left(a^{2} + 6 a + 3\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(7 a^{2} + 12 a + 8\right)\cdot 13^{4} + \left(10 a^{2} + a\right)\cdot 13^{5} + \left(9 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(6 a + 9\right)\cdot 13^{8} + \left(4 a^{2} + a + 3\right)\cdot 13^{9} +O(13^{10})$$ 10*a^2 + 11*a + 10 + (11*a^2 + 11*a + 10)*13 + (a^2 + 6*a + 3)*13^2 + (3*a^2 + 9*a + 2)*13^3 + (7*a^2 + 12*a + 8)*13^4 + (10*a^2 + a)*13^5 + (9*a^2 + 8*a + 1)*13^6 + (a^2 + 8*a + 8)*13^7 + (6*a + 9)*13^8 + (4*a^2 + a + 3)*13^9+O(13^10) $r_{ 3 }$ $=$ $$9 a^{2} + 10 a + 8 + \left(3 a^{2} + 8 a + 10\right)\cdot 13 + \left(a^{2} + 5 a\right)\cdot 13^{2} + \left(11 a^{2} + 9 a + 4\right)\cdot 13^{3} + \left(10 a^{2} + 3\right)\cdot 13^{4} + \left(8 a^{2} + a + 5\right)\cdot 13^{5} + \left(5 a^{2} + 2 a + 5\right)\cdot 13^{6} + \left(8 a^{2} + 5 a\right)\cdot 13^{7} + \left(4 a^{2} + 5\right)\cdot 13^{8} + \left(6 a^{2} + a + 8\right)\cdot 13^{9} +O(13^{10})$$ 9*a^2 + 10*a + 8 + (3*a^2 + 8*a + 10)*13 + (a^2 + 5*a)*13^2 + (11*a^2 + 9*a + 4)*13^3 + (10*a^2 + 3)*13^4 + (8*a^2 + a + 5)*13^5 + (5*a^2 + 2*a + 5)*13^6 + (8*a^2 + 5*a)*13^7 + (4*a^2 + 5)*13^8 + (6*a^2 + a + 8)*13^9+O(13^10) $r_{ 4 }$ $=$ $$11 a^{2} + 6 a + 9 + \left(6 a^{2} + 3 a + 12\right)\cdot 13 + \left(6 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(11 a^{2} + 12 a + 6\right)\cdot 13^{3} + \left(7 a^{2} + 12 a + 1\right)\cdot 13^{4} + \left(8 a^{2} + 6 a + 1\right)\cdot 13^{5} + \left(3 a + 2\right)\cdot 13^{6} + \left(10 a^{2} + 5 a + 1\right)\cdot 13^{7} + \left(5 a^{2} + 8 a + 8\right)\cdot 13^{8} + \left(5 a^{2} + 7 a + 4\right)\cdot 13^{9} +O(13^{10})$$ 11*a^2 + 6*a + 9 + (6*a^2 + 3*a + 12)*13 + (6*a^2 + 7*a + 3)*13^2 + (11*a^2 + 12*a + 6)*13^3 + (7*a^2 + 12*a + 1)*13^4 + (8*a^2 + 6*a + 1)*13^5 + (3*a + 2)*13^6 + (10*a^2 + 5*a + 1)*13^7 + (5*a^2 + 8*a + 8)*13^8 + (5*a^2 + 7*a + 4)*13^9+O(13^10) $r_{ 5 }$ $=$ $$7 a + 9 + \left(11 a^{2} + 7 a + 11\right)\cdot 13 + \left(7 a^{2} + 8 a\right)\cdot 13^{2} + \left(10 a + 3\right)\cdot 13^{3} + \left(a^{2} + 7 a + 3\right)\cdot 13^{4} + \left(5 a^{2} + 3 a\right)\cdot 13^{5} + \left(5 a^{2} + a + 5\right)\cdot 13^{6} + \left(4 a^{2} + 8 a + 12\right)\cdot 13^{7} + \left(7 a^{2} + 2 a + 12\right)\cdot 13^{8} + \left(8 a^{2} + 7 a + 6\right)\cdot 13^{9} +O(13^{10})$$ 7*a + 9 + (11*a^2 + 7*a + 11)*13 + (7*a^2 + 8*a)*13^2 + (10*a + 3)*13^3 + (a^2 + 7*a + 3)*13^4 + (5*a^2 + 3*a)*13^5 + (5*a^2 + a + 5)*13^6 + (4*a^2 + 8*a + 12)*13^7 + (7*a^2 + 2*a + 12)*13^8 + (8*a^2 + 7*a + 6)*13^9+O(13^10) $r_{ 6 }$ $=$ $$8 a^{2} + 2 a + 5 + \left(8 a^{2} + 4 a + 6\right)\cdot 13 + \left(9 a^{2} + 5 a + 12\right)\cdot 13^{2} + \left(2 a^{2} + 6 a + 11\right)\cdot 13^{3} + \left(10 a^{2} + 8\right)\cdot 13^{4} + \left(10 a^{2} + 9 a + 12\right)\cdot 13^{5} + \left(a^{2} + 6 a + 7\right)\cdot 13^{6} + \left(8 a + 9\right)\cdot 13^{7} + \left(12 a^{2} + 11 a + 7\right)\cdot 13^{8} + \left(4 a^{2} + 7 a + 12\right)\cdot 13^{9} +O(13^{10})$$ 8*a^2 + 2*a + 5 + (8*a^2 + 4*a + 6)*13 + (9*a^2 + 5*a + 12)*13^2 + (2*a^2 + 6*a + 11)*13^3 + (10*a^2 + 8)*13^4 + (10*a^2 + 9*a + 12)*13^5 + (a^2 + 6*a + 7)*13^6 + (8*a + 9)*13^7 + (12*a^2 + 11*a + 7)*13^8 + (4*a^2 + 7*a + 12)*13^9+O(13^10) $r_{ 7 }$ $=$ $$a^{2} + 2 a + 11 + \left(6 a^{2} + 12 a + 11\right)\cdot 13 + \left(12 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(8 a^{2} + 1\right)\cdot 13^{3} + \left(a^{2} + 8 a + 5\right)\cdot 13^{4} + \left(3 a^{2} + 8 a + 12\right)\cdot 13^{5} + \left(9 a^{2} + 9 a + 8\right)\cdot 13^{6} + \left(12 a^{2} + 4 a + 9\right)\cdot 13^{7} + \left(12 a^{2} + 4 a\right)\cdot 13^{8} + \left(9 a^{2} + 3\right)\cdot 13^{9} +O(13^{10})$$ a^2 + 2*a + 11 + (6*a^2 + 12*a + 11)*13 + (12*a^2 + 8*a + 4)*13^2 + (8*a^2 + 1)*13^3 + (a^2 + 8*a + 5)*13^4 + (3*a^2 + 8*a + 12)*13^5 + (9*a^2 + 9*a + 8)*13^6 + (12*a^2 + 4*a + 9)*13^7 + (12*a^2 + 4*a)*13^8 + (9*a^2 + 3)*13^9+O(13^10) $r_{ 8 }$ $=$ $$2 a^{2} + 8 + \left(8 a^{2} + 2 a + 1\right)\cdot 13 + \left(11 a^{2} + 10 a + 8\right)\cdot 13^{2} + \left(2 a + 3\right)\cdot 13^{3} + \left(4 a^{2} + 5 a + 8\right)\cdot 13^{4} + \left(12 a^{2} + 2 a + 11\right)\cdot 13^{5} + \left(6 a^{2} + 8 a + 5\right)\cdot 13^{6} + \left(11 a^{2} + 12 a + 12\right)\cdot 13^{7} + \left(12 a^{2} + a + 4\right)\cdot 13^{8} + \left(11 a^{2} + 11 a + 1\right)\cdot 13^{9} +O(13^{10})$$ 2*a^2 + 8 + (8*a^2 + 2*a + 1)*13 + (11*a^2 + 10*a + 8)*13^2 + (2*a + 3)*13^3 + (4*a^2 + 5*a + 8)*13^4 + (12*a^2 + 2*a + 11)*13^5 + (6*a^2 + 8*a + 5)*13^6 + (11*a^2 + 12*a + 12)*13^7 + (12*a^2 + a + 4)*13^8 + (11*a^2 + 11*a + 1)*13^9+O(13^10) $r_{ 9 }$ $=$ $$7 a^{2} + 5 a + 8 + \left(10 a^{2} + 5 a + 4\right)\cdot 13 + \left(9 a^{2} + 8\right)\cdot 13^{2} + \left(11 a^{2} + 7 a + 6\right)\cdot 13^{3} + \left(7 a^{2} + 12 a + 1\right)\cdot 13^{4} + \left(6 a^{2} + 9 a + 7\right)\cdot 13^{5} + \left(10 a^{2} + 2 a + 6\right)\cdot 13^{6} + \left(2 a^{2} + 12 a + 4\right)\cdot 13^{7} + \left(8 a^{2} + 5 a + 11\right)\cdot 13^{8} + \left(2 a^{2} + 10 a\right)\cdot 13^{9} +O(13^{10})$$ 7*a^2 + 5*a + 8 + (10*a^2 + 5*a + 4)*13 + (9*a^2 + 8)*13^2 + (11*a^2 + 7*a + 6)*13^3 + (7*a^2 + 12*a + 1)*13^4 + (6*a^2 + 9*a + 7)*13^5 + (10*a^2 + 2*a + 6)*13^6 + (2*a^2 + 12*a + 4)*13^7 + (8*a^2 + 5*a + 11)*13^8 + (2*a^2 + 10*a)*13^9+O(13^10)

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.