Basic invariants
| Dimension: | $6$ |
| Group: | $C_3^2 : D_{6} $ |
| Conductor: | \(226505550528\)\(\medspace = 2^{6} \cdot 3^{3} \cdot 107^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.2116874304.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T51 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_3^2:D_6$ |
| Projective stem field: | Galois closure of 9.1.2116874304.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{8} + 12x^{7} - 21x^{6} + 28x^{5} - 24x^{4} + 15x^{3} - 6x^{2} + 4x - 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 7 a + \left(6 a^{2} + 5 a + 3\right)\cdot 13 + \left(6 a^{2} + 8 a + 5\right)\cdot 13^{2} + \left(8 a^{2} + 3 a + 12\right)\cdot 13^{3} + \left(11 a^{2} + 6 a + 4\right)\cdot 13^{4} + \left(2 a^{2} + 8 a + 2\right)\cdot 13^{5} + \left(5 a^{2} + a + 7\right)\cdot 13^{6} + \left(3 a^{2} + 11 a + 10\right)\cdot 13^{7} + \left(5 a^{2} + a + 8\right)\cdot 13^{8} + \left(4 a^{2} + 1\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 12 a^{2} + a + 9 + \left(4 a^{2} + 10 a + 9\right)\cdot 13 + \left(11 a + 10\right)\cdot 13^{2} + \left(5 a^{2} + 12 a + 3\right)\cdot 13^{3} + \left(6 a^{2} + 10 a + 4\right)\cdot 13^{4} + \left(3 a + 2\right)\cdot 13^{5} + \left(8 a + 5\right)\cdot 13^{6} + \left(2 a^{2} + 9\right)\cdot 13^{7} + \left(3 a + 3\right)\cdot 13^{8} + \left(4 a^{2} + 8 a + 5\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 a^{2} + 8 a + 7 + \left(7 a^{2} + 7 a + 12\right)\cdot 13 + \left(3 a^{2} + 5 a + 10\right)\cdot 13^{2} + \left(4 a^{2} + 6 a + 11\right)\cdot 13^{3} + \left(a + 10\right)\cdot 13^{4} + \left(5 a^{2} + 5 a + 10\right)\cdot 13^{5} + \left(10 a^{2} + 7 a + 12\right)\cdot 13^{6} + \left(6 a^{2} + 8 a + 4\right)\cdot 13^{7} + \left(9 a^{2} + 7\right)\cdot 13^{8} + \left(6 a^{2} + 7 a + 4\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 10 a^{2} + 2 a + 7 + \left(10 a^{2} + 10 a + 8\right)\cdot 13 + \left(6 a^{2} + 6 a + 5\right)\cdot 13^{2} + \left(11 a^{2} + 8 a + 3\right)\cdot 13^{3} + \left(a + 12\right)\cdot 13^{4} + \left(8 a + 2\right)\cdot 13^{5} + \left(6 a^{2} + 4 a + 8\right)\cdot 13^{6} + \left(9 a^{2} + 12 a + 5\right)\cdot 13^{7} + \left(a^{2} + 11 a + 8\right)\cdot 13^{8} + \left(12 a^{2} + 4 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 8 a^{2} + 8 + \left(2 a^{2} + 11 a + 10\right)\cdot 13 + \left(7 a^{2} + 8 a + 2\right)\cdot 13^{2} + \left(a^{2} + 7 a + 8\right)\cdot 13^{3} + \left(12 a^{2} + 12 a\right)\cdot 13^{4} + \left(a^{2} + 12 a + 11\right)\cdot 13^{5} + \left(6 a^{2} + 9 a + 2\right)\cdot 13^{6} + \left(2 a^{2} + 3 a + 12\right)\cdot 13^{7} + \left(10 a^{2} + 10 a + 3\right)\cdot 13^{8} + \left(8 a^{2} + 5 a + 7\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 8 a^{2} + 4 a + \left(8 a^{2} + 10 a + 10\right)\cdot 13 + \left(12 a^{2} + 10 a + 4\right)\cdot 13^{2} + 5 a^{2} 13^{3} + \left(5 a + 3\right)\cdot 13^{4} + \left(10 a^{2} + 9 a + 3\right)\cdot 13^{5} + \left(a^{2} + 6 a + 11\right)\cdot 13^{6} + \left(2 a + 1\right)\cdot 13^{7} + \left(6 a^{2} + 12 a + 1\right)\cdot 13^{8} + \left(9 a^{2} + 7 a + 4\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( a^{2} + 4 a + 3 + \left(7 a^{2} + 6 a + 8\right)\cdot 13 + \left(4 a^{2} + a + 7\right)\cdot 13^{2} + \left(11 a^{2} + 5 a + 3\right)\cdot 13^{3} + \left(9 a^{2} + 12 a\right)\cdot 13^{4} + \left(11 a^{2} + 9 a\right)\cdot 13^{5} + \left(9 a^{2} + 1\right)\cdot 13^{6} + \left(5 a^{2} + 8 a + 10\right)\cdot 13^{7} + \left(8 a + 12\right)\cdot 13^{8} + \left(12 a^{2} + 11 a + 2\right)\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( a^{2} + 5 a + 3 + \left(3 a^{2} + 7 a + 11\right)\cdot 13 + \left(2 a^{2} + 11 a + 4\right)\cdot 13^{2} + \left(7 a^{2} + 11 a + 11\right)\cdot 13^{3} + \left(11 a + 6\right)\cdot 13^{4} + \left(6 a^{2} + 7 a + 3\right)\cdot 13^{5} + \left(9 a^{2} + 8 a + 7\right)\cdot 13^{6} + \left(3 a^{2} + 9\right)\cdot 13^{7} + \left(6 a^{2} + 2 a + 11\right)\cdot 13^{8} + 10 a^{2} 13^{9} +O(13^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 8 a + 6 + \left(a^{2} + 9 a + 4\right)\cdot 13 + \left(8 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(9 a^{2} + 7 a + 9\right)\cdot 13^{3} + \left(9 a^{2} + 2 a + 8\right)\cdot 13^{4} + \left(12 a + 2\right)\cdot 13^{5} + \left(3 a^{2} + 3 a + 9\right)\cdot 13^{6} + \left(5 a^{2} + 4 a\right)\cdot 13^{7} + \left(12 a^{2} + a + 7\right)\cdot 13^{8} + \left(9 a^{2} + 6 a + 4\right)\cdot 13^{9} +O(13^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,3)(2,5)(6,8)$ | $0$ |
| $9$ | $2$ | $(2,6)(4,8)(5,9)$ | $-2$ |
| $9$ | $2$ | $(1,9)(3,8)(4,5)(6,7)$ | $0$ |
| $2$ | $3$ | $(1,7,3)(2,5,4)(6,9,8)$ | $-3$ |
| $6$ | $3$ | $(1,5,6)(2,8,3)(4,9,7)$ | $0$ |
| $6$ | $3$ | $(1,7,3)(6,8,9)$ | $0$ |
| $12$ | $3$ | $(1,6,2)(3,8,4)(5,7,9)$ | $0$ |
| $18$ | $6$ | $(1,8,5,3,6,2)(4,7,9)$ | $0$ |
| $18$ | $6$ | $(1,7,3)(2,9,4,6,5,8)$ | $1$ |
| $18$ | $6$ | $(1,8,7,9,3,6)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.