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
| $f(x)$ | $=$ |
\( x^{9} - 7x^{7} - 7x^{6} - 7x^{5} + 84x^{4} + 35x^{3} - 280x^{2} + 28x + 280 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 4 a^{2} + 9 a + 10 + \left(11 a^{2} + 9 a + 7\right)\cdot 13 + \left(3 a^{2} + 11 a + 8\right)\cdot 13^{2} + \left(a^{2} + 5 a + 12\right)\cdot 13^{3} + \left(a^{2} + 4 a + 11\right)\cdot 13^{4} + \left(12 a^{2} + 8 a\right)\cdot 13^{5} + \left(a^{2} + 9 a + 9\right)\cdot 13^{6} + \left(12 a + 6\right)\cdot 13^{7} + \left(a^{2} + 9 a + 4\right)\cdot 13^{8} + \left(11 a^{2} + 4 a + 10\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 10 a^{2} + 11 a + 10 + \left(11 a^{2} + 11 a + 10\right)\cdot 13 + \left(a^{2} + 6 a + 3\right)\cdot 13^{2} + \left(3 a^{2} + 9 a + 2\right)\cdot 13^{3} + \left(7 a^{2} + 12 a + 8\right)\cdot 13^{4} + \left(10 a^{2} + a\right)\cdot 13^{5} + \left(9 a^{2} + 8 a + 1\right)\cdot 13^{6} + \left(a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(6 a + 9\right)\cdot 13^{8} + \left(4 a^{2} + a + 3\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 9 a^{2} + 10 a + 8 + \left(3 a^{2} + 8 a + 10\right)\cdot 13 + \left(a^{2} + 5 a\right)\cdot 13^{2} + \left(11 a^{2} + 9 a + 4\right)\cdot 13^{3} + \left(10 a^{2} + 3\right)\cdot 13^{4} + \left(8 a^{2} + a + 5\right)\cdot 13^{5} + \left(5 a^{2} + 2 a + 5\right)\cdot 13^{6} + \left(8 a^{2} + 5 a\right)\cdot 13^{7} + \left(4 a^{2} + 5\right)\cdot 13^{8} + \left(6 a^{2} + a + 8\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 11 a^{2} + 6 a + 9 + \left(6 a^{2} + 3 a + 12\right)\cdot 13 + \left(6 a^{2} + 7 a + 3\right)\cdot 13^{2} + \left(11 a^{2} + 12 a + 6\right)\cdot 13^{3} + \left(7 a^{2} + 12 a + 1\right)\cdot 13^{4} + \left(8 a^{2} + 6 a + 1\right)\cdot 13^{5} + \left(3 a + 2\right)\cdot 13^{6} + \left(10 a^{2} + 5 a + 1\right)\cdot 13^{7} + \left(5 a^{2} + 8 a + 8\right)\cdot 13^{8} + \left(5 a^{2} + 7 a + 4\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 7 a + 9 + \left(11 a^{2} + 7 a + 11\right)\cdot 13 + \left(7 a^{2} + 8 a\right)\cdot 13^{2} + \left(10 a + 3\right)\cdot 13^{3} + \left(a^{2} + 7 a + 3\right)\cdot 13^{4} + \left(5 a^{2} + 3 a\right)\cdot 13^{5} + \left(5 a^{2} + a + 5\right)\cdot 13^{6} + \left(4 a^{2} + 8 a + 12\right)\cdot 13^{7} + \left(7 a^{2} + 2 a + 12\right)\cdot 13^{8} + \left(8 a^{2} + 7 a + 6\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 8 a^{2} + 2 a + 5 + \left(8 a^{2} + 4 a + 6\right)\cdot 13 + \left(9 a^{2} + 5 a + 12\right)\cdot 13^{2} + \left(2 a^{2} + 6 a + 11\right)\cdot 13^{3} + \left(10 a^{2} + 8\right)\cdot 13^{4} + \left(10 a^{2} + 9 a + 12\right)\cdot 13^{5} + \left(a^{2} + 6 a + 7\right)\cdot 13^{6} + \left(8 a + 9\right)\cdot 13^{7} + \left(12 a^{2} + 11 a + 7\right)\cdot 13^{8} + \left(4 a^{2} + 7 a + 12\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 7 }$ |
$=$ |
\( a^{2} + 2 a + 11 + \left(6 a^{2} + 12 a + 11\right)\cdot 13 + \left(12 a^{2} + 8 a + 4\right)\cdot 13^{2} + \left(8 a^{2} + 1\right)\cdot 13^{3} + \left(a^{2} + 8 a + 5\right)\cdot 13^{4} + \left(3 a^{2} + 8 a + 12\right)\cdot 13^{5} + \left(9 a^{2} + 9 a + 8\right)\cdot 13^{6} + \left(12 a^{2} + 4 a + 9\right)\cdot 13^{7} + \left(12 a^{2} + 4 a\right)\cdot 13^{8} + \left(9 a^{2} + 3\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 2 a^{2} + 8 + \left(8 a^{2} + 2 a + 1\right)\cdot 13 + \left(11 a^{2} + 10 a + 8\right)\cdot 13^{2} + \left(2 a + 3\right)\cdot 13^{3} + \left(4 a^{2} + 5 a + 8\right)\cdot 13^{4} + \left(12 a^{2} + 2 a + 11\right)\cdot 13^{5} + \left(6 a^{2} + 8 a + 5\right)\cdot 13^{6} + \left(11 a^{2} + 12 a + 12\right)\cdot 13^{7} + \left(12 a^{2} + a + 4\right)\cdot 13^{8} + \left(11 a^{2} + 11 a + 1\right)\cdot 13^{9} +O(13^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 7 a^{2} + 5 a + 8 + \left(10 a^{2} + 5 a + 4\right)\cdot 13 + \left(9 a^{2} + 8\right)\cdot 13^{2} + \left(11 a^{2} + 7 a + 6\right)\cdot 13^{3} + \left(7 a^{2} + 12 a + 1\right)\cdot 13^{4} + \left(6 a^{2} + 9 a + 7\right)\cdot 13^{5} + \left(10 a^{2} + 2 a + 6\right)\cdot 13^{6} + \left(2 a^{2} + 12 a + 4\right)\cdot 13^{7} + \left(8 a^{2} + 5 a + 11\right)\cdot 13^{8} + \left(2 a^{2} + 10 a\right)\cdot 13^{9} +O(13^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,6)(4,5)$ | $0$ |
| $54$ | $2$ | $(1,6)(2,4)(3,5)(8,9)$ | $2$ |
| $6$ | $3$ | $(1,7,6)$ | $0$ |
| $8$ | $3$ | $(1,6,7)(2,3,9)(4,5,8)$ | $3$ |
| $12$ | $3$ | $(1,7,6)(4,8,5)$ | $-3$ |
| $72$ | $3$ | $(1,2,4)(3,5,6)(7,9,8)$ | $0$ |
| $54$ | $4$ | $(1,5,6,4)(7,8)$ | $0$ |
| $54$ | $6$ | $(1,6,7)(2,3)(4,5)$ | $0$ |
| $108$ | $6$ | $(1,8,7,5,6,4)(2,3)$ | $-1$ |
| $72$ | $9$ | $(1,3,5,6,9,8,7,2,4)$ | $0$ |
| $72$ | $9$ | $(1,9,8,7,3,5,6,2,4)$ | $0$ |
| $54$ | $12$ | $(1,7,6)(2,5,3,4)(8,9)$ | $0$ |
| $54$ | $12$ | $(1,6,7)(2,5,3,4)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.