Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(7319366412146449\)\(\medspace = 1489^{5}\) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.4915625528641.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | even |
| Determinant: | 1.1489.2t1.a.a |
| Projective image: | $C_3^3.S_4$ |
| Projective stem field: | Galois closure of 9.5.4915625528641.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} - x^{7} + 11x^{6} - 25x^{5} - 33x^{4} + 35x^{3} + 18x^{2} + 14x + 29 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 61 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 61 }$:
\( x^{3} + 7x + 59 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 6 a^{2} + 7 a + 24 + \left(6 a^{2} + 47 a + 51\right)\cdot 61 + \left(31 a^{2} + 30 a\right)\cdot 61^{2} + \left(32 a^{2} + 28 a + 53\right)\cdot 61^{3} + \left(43 a^{2} + 40 a + 11\right)\cdot 61^{4} + \left(25 a^{2} + 4 a + 38\right)\cdot 61^{5} + \left(43 a^{2} + 17 a + 56\right)\cdot 61^{6} + \left(16 a^{2} + a + 60\right)\cdot 61^{7} + \left(50 a^{2} + 7 a + 11\right)\cdot 61^{8} + \left(59 a^{2} + 34 a + 3\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 56 a^{2} + 33 a + 33 + \left(6 a^{2} + 36 a + 20\right)\cdot 61 + \left(37 a^{2} + 5 a + 22\right)\cdot 61^{2} + \left(47 a^{2} + 51 a + 59\right)\cdot 61^{3} + \left(44 a^{2} + 14 a + 44\right)\cdot 61^{4} + \left(50 a^{2} + 48 a + 46\right)\cdot 61^{5} + \left(19 a^{2} + 16 a + 57\right)\cdot 61^{6} + \left(33 a^{2} + a + 24\right)\cdot 61^{7} + \left(a^{2} + 29 a + 33\right)\cdot 61^{8} + \left(9 a^{2} + 31 a + 21\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 24 a^{2} + 48 a + 47 + \left(47 a^{2} + a + 40\right)\cdot 61 + \left(37 a^{2} + 8 a + 11\right)\cdot 61^{2} + \left(27 a^{2} + 53 a + 30\right)\cdot 61^{3} + \left(46 a^{2} + 2 a + 25\right)\cdot 61^{4} + \left(59 a^{2} + 50 a + 34\right)\cdot 61^{5} + \left(39 a^{2} + 6 a + 40\right)\cdot 61^{6} + \left(7 a^{2} + 37 a + 18\right)\cdot 61^{7} + \left(26 a^{2} + 3 a + 21\right)\cdot 61^{8} + \left(7 a^{2} + 22 a + 43\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 47 a^{2} + 54 a + 52 + \left(7 a^{2} + 32 a + 44\right)\cdot 61 + \left(36 a^{2} + 43 a + 17\right)\cdot 61^{2} + \left(41 a^{2} + 22 a + 31\right)\cdot 61^{3} + \left(42 a^{2} + 54 a + 55\right)\cdot 61^{4} + \left(38 a^{2} + 5 a + 10\right)\cdot 61^{5} + \left(23 a^{2} + 46 a + 55\right)\cdot 61^{6} + \left(34 a^{2} + 27 a + 29\right)\cdot 61^{7} + \left(60 a^{2} + 15 a + 44\right)\cdot 61^{8} + \left(2 a^{2} + 29 a + 13\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 19 a^{2} + 35 a + 23 + \left(46 a^{2} + 52 a + 21\right)\cdot 61 + \left(48 a^{2} + 11 a + 56\right)\cdot 61^{2} + \left(32 a^{2} + 48 a + 30\right)\cdot 61^{3} + \left(34 a^{2} + 52 a + 17\right)\cdot 61^{4} + \left(32 a^{2} + 6 a + 43\right)\cdot 61^{5} + \left(17 a^{2} + 59 a + 26\right)\cdot 61^{6} + \left(54 a^{2} + 31 a + 41\right)\cdot 61^{7} + \left(59 a^{2} + 16 a\right)\cdot 61^{8} + \left(48 a^{2} + 25\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 29 + 5\cdot 61 + 47\cdot 61^{2} + 60\cdot 61^{3} + 20\cdot 61^{4} + 48\cdot 61^{5} + 42\cdot 61^{6} + 55\cdot 61^{7} + 27\cdot 61^{8} + 2\cdot 61^{9} +O(61^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 39 + 58\cdot 61 + 41\cdot 61^{2} + 20\cdot 61^{3} + 37\cdot 61^{4} + 50\cdot 61^{5} + 3\cdot 61^{6} + 50\cdot 61^{7} + 17\cdot 61^{8} + 56\cdot 61^{9} +O(61^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 31 a^{2} + 6 a + 39 + \left(7 a^{2} + 12 a + 37\right)\cdot 61 + \left(53 a^{2} + 22 a + 42\right)\cdot 61^{2} + \left(40 a + 47\right)\cdot 61^{3} + \left(32 a^{2} + 17 a + 59\right)\cdot 61^{4} + \left(36 a^{2} + 6 a + 47\right)\cdot 61^{5} + \left(38 a^{2} + 37 a + 13\right)\cdot 61^{6} + \left(36 a^{2} + 22 a + 52\right)\cdot 61^{7} + \left(45 a^{2} + 50 a + 30\right)\cdot 61^{8} + \left(54 a^{2} + 4 a + 40\right)\cdot 61^{9} +O(61^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 21 + 24\cdot 61 + 3\cdot 61^{2} + 32\cdot 61^{3} + 31\cdot 61^{4} + 45\cdot 61^{5} + 7\cdot 61^{6} + 32\cdot 61^{7} + 55\cdot 61^{8} + 37\cdot 61^{9} +O(61^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,3)(6,7)$ | $0$ |
| $54$ | $2$ | $(1,6)(2,4)(3,7)(8,9)$ | $2$ |
| $6$ | $3$ | $(6,7,9)$ | $0$ |
| $8$ | $3$ | $(1,3,8)(2,4,5)(6,7,9)$ | $3$ |
| $12$ | $3$ | $(1,3,8)(6,7,9)$ | $-3$ |
| $72$ | $3$ | $(1,6,2)(3,7,4)(5,8,9)$ | $0$ |
| $54$ | $4$ | $(1,6,3,7)(8,9)$ | $0$ |
| $54$ | $6$ | $(1,3)(2,4)(6,9,7)$ | $0$ |
| $108$ | $6$ | $(1,6,8,9,3,7)(2,4)$ | $-1$ |
| $72$ | $9$ | $(1,6,4,3,7,5,8,9,2)$ | $0$ |
| $72$ | $9$ | $(1,6,5,8,9,4,3,7,2)$ | $0$ |
| $54$ | $12$ | $(1,2,3,4)(5,8)(6,7,9)$ | $0$ |
| $54$ | $12$ | $(1,2,3,4)(5,8)(6,9,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.