Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(385129317228262144\)\(\medspace = 2^{8} \cdot 19^{5} \cdot 157^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.1309465408.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Determinant: | 1.19.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.1309465408.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} + x^{7} + 2x^{5} - 5x^{4} + 5x^{3} - x^{2} - 2x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 2 a^{2} + a + 9 + 8 a\cdot 11 + \left(4 a^{2} + 7 a + 2\right)\cdot 11^{2} + \left(3 a^{2} + 3 a\right)\cdot 11^{3} + \left(8 a^{2} + 7 a + 1\right)\cdot 11^{4} + \left(9 a^{2} + 8 a + 2\right)\cdot 11^{5} + \left(a + 7\right)\cdot 11^{6} + \left(8 a^{2} + a + 1\right)\cdot 11^{7} + \left(4 a^{2} + 8 a + 8\right)\cdot 11^{8} + \left(8 a^{2} + 7 a + 6\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 10 a^{2} + 7 a + 4 + \left(5 a^{2} + 2 a + 6\right)\cdot 11 + \left(2 a^{2} + 5 a + 4\right)\cdot 11^{2} + \left(8 a^{2} + 7 a + 6\right)\cdot 11^{3} + \left(3 a^{2} + 1\right)\cdot 11^{4} + \left(4 a^{2} + 3 a\right)\cdot 11^{5} + \left(7 a^{2} + 5 a\right)\cdot 11^{6} + \left(a^{2} + 3\right)\cdot 11^{7} + \left(10 a^{2} + 7 a + 10\right)\cdot 11^{8} + \left(10 a^{2} + 9 a + 7\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 7 a^{2} + 6 a + 9 + \left(4 a^{2} + 8 a + 10\right)\cdot 11 + \left(2 a^{2} + 10 a + 8\right)\cdot 11^{2} + \left(2 a^{2} + 5 a\right)\cdot 11^{3} + \left(2 a^{2} + 6 a + 9\right)\cdot 11^{4} + \left(2 a^{2} + 5 a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 6 a + 1\right)\cdot 11^{6} + \left(4 a^{2} + 3 a\right)\cdot 11^{7} + \left(8 a^{2} + a + 9\right)\cdot 11^{8} + \left(3 a^{2} + 7 a + 8\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 4 }$ |
$=$ |
\( a^{2} + 4 a + 4 + \left(2 a^{2} + 4 a + 3\right)\cdot 11 + \left(2 a^{2} + 2 a + 3\right)\cdot 11^{2} + \left(10 a^{2} + 5 a + 9\right)\cdot 11^{3} + \left(2 a + 5\right)\cdot 11^{4} + \left(5 a^{2} + 10\right)\cdot 11^{5} + \left(2 a^{2} + 8 a + 1\right)\cdot 11^{6} + \left(3 a^{2} + 5 a + 6\right)\cdot 11^{7} + \left(10 a^{2} + 4\right)\cdot 11^{8} + \left(5 a^{2} + 7 a + 3\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 9 a^{2} + 10 a + 10 + \left(a^{2} + 7 a\right)\cdot 11 + \left(7 a^{2} + 7 a + 7\right)\cdot 11^{2} + \left(9 a^{2} + 5 a + 4\right)\cdot 11^{3} + \left(3 a^{2} + 7 a + 5\right)\cdot 11^{4} + \left(10 a^{2} + 2 a + 4\right)\cdot 11^{5} + \left(9 a^{2} + 10 a + 3\right)\cdot 11^{6} + \left(3 a^{2} + 8 a + 2\right)\cdot 11^{7} + \left(5 a^{2} + 7 a\right)\cdot 11^{8} + \left(5 a^{2} + 9 a + 8\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 3 a^{2} + 5 a + 2 + \left(3 a^{2} + 10\right)\cdot 11 + \left(a^{2} + 9 a + 2\right)\cdot 11^{2} + \left(4 a^{2} + 8 a + 8\right)\cdot 11^{3} + \left(3 a^{2} + 2 a + 4\right)\cdot 11^{4} + \left(7 a^{2} + 5 a\right)\cdot 11^{5} + \left(4 a^{2} + 6 a\right)\cdot 11^{6} + \left(5 a^{2} + a + 8\right)\cdot 11^{7} + \left(6 a^{2} + 7 a + 1\right)\cdot 11^{8} + \left(5 a^{2} + 2 a + 8\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 7 }$ |
$=$ |
\( a^{2} + 9 a + 1 + \left(7 a^{2} + 3 a + 3\right)\cdot 11 + \left(5 a^{2} + 6 a + 2\right)\cdot 11^{2} + \left(10 a^{2} + 6 a + 8\right)\cdot 11^{3} + \left(2 a^{2} + 2 a + 2\right)\cdot 11^{4} + \left(5 a^{2} + 6 a + 5\right)\cdot 11^{5} + \left(10 a^{2} + 9 a + 10\right)\cdot 11^{6} + \left(7 a^{2} + 9 a + 7\right)\cdot 11^{7} + \left(4 a^{2} + 9 a + 7\right)\cdot 11^{8} + \left(a^{2} + 5\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 3 a^{2} + 7 a + \left(10 a^{2} + 9 a\right)\cdot 11 + \left(2 a^{2} + 4 a + 6\right)\cdot 11^{2} + \left(9 a^{2} + 9 a + 6\right)\cdot 11^{3} + \left(5 a^{2} + a + 6\right)\cdot 11^{4} + \left(3 a^{2} + 10 a + 6\right)\cdot 11^{5} + \left(2 a^{2} + 5 a + 10\right)\cdot 11^{6} + \left(9 a^{2} + 8 a + 5\right)\cdot 11^{7} + \left(8 a^{2} + 10 a + 9\right)\cdot 11^{8} + \left(5 a^{2} + 2 a + 7\right)\cdot 11^{9} +O(11^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 8 a^{2} + 6 a + 6 + \left(8 a^{2} + 9 a + 8\right)\cdot 11 + \left(4 a^{2} + 6\right)\cdot 11^{2} + \left(8 a^{2} + 2 a + 10\right)\cdot 11^{3} + \left(a^{2} + a + 6\right)\cdot 11^{4} + \left(7 a^{2} + 2 a + 9\right)\cdot 11^{5} + \left(7 a^{2} + a + 8\right)\cdot 11^{6} + \left(10 a^{2} + 4 a + 8\right)\cdot 11^{7} + \left(6 a^{2} + 2 a + 3\right)\cdot 11^{8} + \left(7 a^{2} + 7 a + 9\right)\cdot 11^{9} +O(11^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(3,4)$ | $4$ |
| $18$ | $2$ | $(3,5)(4,7)(6,9)$ | $2$ |
| $27$ | $2$ | $(1,2)(3,4)(5,7)$ | $0$ |
| $27$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $2$ |
| $6$ | $3$ | $(1,2,8)$ | $0$ |
| $8$ | $3$ | $(1,8,2)(3,6,4)(5,9,7)$ | $3$ |
| $12$ | $3$ | $(1,8,2)(3,6,4)$ | $-3$ |
| $72$ | $3$ | $(1,3,5)(2,4,7)(6,9,8)$ | $0$ |
| $54$ | $4$ | $(1,4,2,3)(6,8)$ | $0$ |
| $162$ | $4$ | $(1,4,2,3)(5,7)(6,8)$ | $0$ |
| $36$ | $6$ | $(1,2,8)(3,5)(4,7)(6,9)$ | $2$ |
| $36$ | $6$ | $(1,6,8,4,2,3)$ | $-1$ |
| $36$ | $6$ | $(1,2,8)(3,4)$ | $-2$ |
| $36$ | $6$ | $(1,2,8)(3,4)(5,7,9)$ | $1$ |
| $54$ | $6$ | $(1,8,2)(3,4)(5,7)$ | $0$ |
| $72$ | $6$ | $(1,2,8)(3,5,6,9,4,7)$ | $-1$ |
| $108$ | $6$ | $(1,6,8,4,2,3)(5,7)$ | $-1$ |
| $216$ | $6$ | $(1,4,7,2,3,5)(6,9,8)$ | $0$ |
| $144$ | $9$ | $(1,6,9,8,4,7,2,3,5)$ | $0$ |
| $108$ | $12$ | $(1,2,8)(3,5,4,7)(6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.