Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(219708209445758929\)\(\medspace = 7^{10} \cdot 167^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.962721767.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr C_3$ |
| Projective stem field: | Galois closure of 9.3.962721767.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} + 2x^{7} - 4x^{2} + 4x - 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 9 a^{2} + 7 a + 6 + \left(6 a^{2} + 2 a + 1\right)\cdot 11 + \left(10 a + 5\right)\cdot 11^{2} + \left(10 a^{2} + 9 a + 2\right)\cdot 11^{3} + \left(7 a^{2} + 2 a + 4\right)\cdot 11^{4} + \left(a^{2} + 9\right)\cdot 11^{5} + \left(6 a + 4\right)\cdot 11^{6} + \left(2 a^{2} + 10 a + 8\right)\cdot 11^{7} + \left(8 a^{2} + 5 a + 5\right)\cdot 11^{8} + \left(9 a^{2} + 6 a + 10\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( a^{2} + 5 a + 9 + \left(2 a^{2} + 3 a + 7\right)\cdot 11 + \left(4 a + 2\right)\cdot 11^{2} + \left(9 a^{2} + 7 a + 10\right)\cdot 11^{3} + \left(8 a^{2} + 9 a + 5\right)\cdot 11^{4} + \left(3 a^{2} + 9 a + 9\right)\cdot 11^{5} + \left(3 a^{2} + 5 a + 9\right)\cdot 11^{6} + \left(3 a^{2} + 2 a + 6\right)\cdot 11^{7} + \left(5 a^{2} + 10 a + 8\right)\cdot 11^{8} + \left(10 a + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 2 a + 5 + \left(a^{2} + 10 a + 8\right)\cdot 11 + a\cdot 11^{2} + \left(2 a^{2} + 10\right)\cdot 11^{3} + \left(4 a^{2} + a + 2\right)\cdot 11^{4} + \left(8 a^{2} + 6 a + 7\right)\cdot 11^{5} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{6} + \left(3 a^{2} + 10 a + 6\right)\cdot 11^{7} + \left(9 a^{2} + 3\right)\cdot 11^{8} + \left(9 a^{2} + 2 a + 3\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 3 a^{2} + 2 a + 8 + \left(2 a^{2} + 8 a\right)\cdot 11 + \left(2 a^{2} + 10 a + 9\right)\cdot 11^{2} + \left(9 a^{2} + 8 a + 6\right)\cdot 11^{3} + \left(6 a^{2} + 10\right)\cdot 11^{4} + \left(7 a^{2} + 8 a + 10\right)\cdot 11^{5} + \left(2 a^{2} + 10 a + 8\right)\cdot 11^{6} + \left(9 a^{2} + 8 a + 3\right)\cdot 11^{7} + \left(5 a^{2} + 4 a + 9\right)\cdot 11^{8} + \left(2 a^{2} + 7 a + 8\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a^{2} + 2 a + 4 + \left(3 a^{2} + 9 a\right)\cdot 11 + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{2} + \left(9 a^{2} + 9\right)\cdot 11^{3} + \left(9 a^{2} + 7 a + 6\right)\cdot 11^{4} + \left(4 a + 4\right)\cdot 11^{5} + \left(5 a^{2} + 6 a\right)\cdot 11^{6} + \left(5 a^{2} + 2\right)\cdot 11^{7} + \left(4 a^{2} + 4 a + 8\right)\cdot 11^{8} + \left(2 a^{2} + 2 a\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 2 a^{2} + 3 a + 5 + 6\cdot 11 + \left(6 a^{2} + 10 a + 8\right)\cdot 11^{2} + \left(a^{2} + 8 a + 3\right)\cdot 11^{3} + \left(a^{2} + a + 6\right)\cdot 11^{4} + \left(4 a^{2} + 2 a + 8\right)\cdot 11^{5} + \left(5 a^{2} + 3 a\right)\cdot 11^{6} + \left(9 a + 7\right)\cdot 11^{7} + \left(6 a^{2} + 8 a\right)\cdot 11^{8} + \left(10 a^{2} + 9 a\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 a^{2} + 1 + \left(3 a^{2} + 7 a + 4\right)\cdot 11 + \left(2 a^{2} + 8 a + 7\right)\cdot 11^{2} + \left(3 a^{2} + 8 a + 9\right)\cdot 11^{3} + \left(6 a^{2} + 8 a + 5\right)\cdot 11^{4} + \left(2 a^{2} + a + 6\right)\cdot 11^{5} + \left(6 a^{2} + 9 a + 5\right)\cdot 11^{6} + \left(3 a^{2} + 7\right)\cdot 11^{7} + \left(7 a^{2} + 10 a + 9\right)\cdot 11^{8} + \left(9 a^{2} + 2 a + 9\right)\cdot 11^{9} +O(11^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 7 a^{2} + 4 a + 6 + \left(6 a^{2} + 10 a + 6\right)\cdot 11 + \left(8 a^{2} + 6 a + 6\right)\cdot 11^{2} + \left(3 a^{2} + 5 a + 10\right)\cdot 11^{3} + \left(6 a^{2} + 9\right)\cdot 11^{4} + \left(10 a^{2} + 4 a + 3\right)\cdot 11^{5} + \left(4 a^{2} + 5 a + 8\right)\cdot 11^{6} + \left(9 a^{2} + 10 a + 7\right)\cdot 11^{7} + \left(10 a^{2} + 6 a + 8\right)\cdot 11^{8} + \left(7 a^{2} + 3 a + 8\right)\cdot 11^{9} +O(11^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 10 a^{2} + 8 a + 1 + \left(6 a^{2} + 3 a + 8\right)\cdot 11 + \left(2 a^{2} + 3 a + 7\right)\cdot 11^{2} + \left(6 a^{2} + 4 a + 2\right)\cdot 11^{3} + \left(3 a^{2} + 2\right)\cdot 11^{4} + \left(4 a^{2} + 7 a + 5\right)\cdot 11^{5} + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{6} + \left(6 a^{2} + 4\right)\cdot 11^{7} + \left(8 a^{2} + 3 a\right)\cdot 11^{8} + \left(a^{2} + 9 a + 3\right)\cdot 11^{9} +O(11^{10})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $12$ | |
| $9$ | $2$ | $(1,2)$ | $4$ | |
| $27$ | $2$ | $(1,2)(3,4)(5,8)$ | $0$ | ✓ |
| $27$ | $2$ | $(1,2)(8,9)$ | $0$ | |
| $6$ | $3$ | $(3,4,6)$ | $0$ | |
| $8$ | $3$ | $(1,2,7)(3,4,6)(5,8,9)$ | $3$ | |
| $12$ | $3$ | $(3,4,6)(5,8,9)$ | $-3$ | |
| $36$ | $3$ | $(1,5,3)(2,8,4)(6,7,9)$ | $0$ | |
| $36$ | $3$ | $(1,3,5)(2,4,8)(6,9,7)$ | $0$ | |
| $18$ | $6$ | $(1,2)(3,4,6)$ | $-2$ | |
| $18$ | $6$ | $(1,2)(5,8,9)$ | $-2$ | |
| $36$ | $6$ | $(1,2)(3,4,6)(5,8,9)$ | $1$ | |
| $54$ | $6$ | $(1,2)(3,4,6)(8,9)$ | $0$ | |
| $108$ | $6$ | $(1,8,4,2,5,3)(6,7,9)$ | $0$ | |
| $108$ | $6$ | $(1,3,5,2,4,8)(6,9,7)$ | $0$ | |
| $72$ | $9$ | $(1,5,3,2,8,4,7,9,6)$ | $0$ | |
| $72$ | $9$ | $(1,3,8,7,6,5,2,4,9)$ | $0$ |