Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(49204012094098099\)\(\medspace = 7^{6} \cdot 211^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.97123852609.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | odd |
| Determinant: | 1.211.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.1.97123852609.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{8} + 7x^{7} - 9x^{6} + 8x^{5} - 21x^{4} + 14x^{3} - 15x^{2} + 14x - 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{3} + 2x + 68 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 a^{2} + 56 a + 71 + \left(11 a^{2} + 46 a + 24\right)\cdot 73 + \left(8 a^{2} + 56 a + 11\right)\cdot 73^{2} + \left(24 a^{2} + 17 a + 41\right)\cdot 73^{3} + \left(35 a^{2} + 11 a + 12\right)\cdot 73^{4} + \left(15 a^{2} + 5 a + 18\right)\cdot 73^{5} + \left(58 a^{2} + 57 a + 21\right)\cdot 73^{6} + \left(37 a^{2} + 49 a + 33\right)\cdot 73^{7} + \left(36 a^{2} + 20 a + 54\right)\cdot 73^{8} + \left(16 a^{2} + 38 a + 1\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 47 a^{2} + 20 a + 22 + \left(64 a^{2} + 14 a + 68\right)\cdot 73 + \left(39 a^{2} + 5 a + 60\right)\cdot 73^{2} + \left(9 a^{2} + 60 a + 14\right)\cdot 73^{3} + \left(9 a^{2} + 43 a + 20\right)\cdot 73^{4} + \left(16 a^{2} + 23 a + 59\right)\cdot 73^{5} + \left(46 a^{2} + 54 a + 23\right)\cdot 73^{6} + \left(27 a^{2} + 46 a + 55\right)\cdot 73^{7} + \left(29 a^{2} + 9 a + 24\right)\cdot 73^{8} + \left(57 a^{2} + 63 a + 9\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 18 + 22\cdot 73 + 9\cdot 73^{2} + 70\cdot 73^{3} + 48\cdot 73^{4} + 39\cdot 73^{5} + 70\cdot 73^{6} + 69\cdot 73^{7} + 36\cdot 73^{8} + 8\cdot 73^{9} +O(73^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 56 a^{2} + 11 a + 34 + \left(47 a^{2} + 42 a + 21\right)\cdot 73 + \left(51 a^{2} + 68 a + 52\right)\cdot 73^{2} + \left(2 a^{2} + 20 a + 5\right)\cdot 73^{3} + \left(68 a^{2} + 63 a + 50\right)\cdot 73^{4} + \left(49 a^{2} + 14 a + 55\right)\cdot 73^{5} + \left(50 a^{2} + 69 a + 29\right)\cdot 73^{6} + \left(57 a^{2} + 24 a + 22\right)\cdot 73^{7} + \left(64 a^{2} + 43 a + 23\right)\cdot 73^{8} + \left(41 a^{2} + 54 a + 37\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 7 + 23\cdot 73 + 30\cdot 73^{2} + 25\cdot 73^{3} + 29\cdot 73^{4} + 30\cdot 73^{5} + 12\cdot 73^{6} + 41\cdot 73^{7} + 37\cdot 73^{8} + 16\cdot 73^{9} +O(73^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 57 a^{2} + 13 a + 39 + \left(18 a^{2} + 5 a + 59\right)\cdot 73 + \left(20 a^{2} + 67 a + 51\right)\cdot 73^{2} + \left(65 a^{2} + 18 a + 71\right)\cdot 73^{3} + \left(28 a^{2} + 72 a + 3\right)\cdot 73^{4} + \left(28 a^{2} + 24 a + 11\right)\cdot 73^{5} + \left(59 a^{2} + 12 a + 47\right)\cdot 73^{6} + \left(49 a^{2} + 38 a\right)\cdot 73^{7} + \left(48 a^{2} + 57 a + 22\right)\cdot 73^{8} + \left(7 a^{2} + 65 a + 14\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 43 a^{2} + 42 a + 41 + \left(33 a^{2} + 16 a + 2\right)\cdot 73 + \left(54 a^{2} + 72 a + 56\right)\cdot 73^{2} + \left(60 a^{2} + 64 a + 58\right)\cdot 73^{3} + \left(68 a^{2} + 38 a + 26\right)\cdot 73^{4} + \left(6 a^{2} + 34 a + 71\right)\cdot 73^{5} + \left(49 a^{2} + 22 a + 51\right)\cdot 73^{6} + \left(60 a^{2} + a + 50\right)\cdot 73^{7} + \left(51 a^{2} + 20 a + 54\right)\cdot 73^{8} + \left(46 a^{2} + 28 a + 43\right)\cdot 73^{9} +O(73^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 66 + 50\cdot 73 + 8\cdot 73^{2} + 17\cdot 73^{3} + 73^{4} + 43\cdot 73^{5} + 53\cdot 73^{6} + 30\cdot 73^{7} + 24\cdot 73^{8} + 18\cdot 73^{9} +O(73^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 8 a^{2} + 4 a + 71 + \left(43 a^{2} + 21 a + 18\right)\cdot 73 + \left(44 a^{2} + 22 a + 11\right)\cdot 73^{2} + \left(56 a^{2} + 36 a + 60\right)\cdot 73^{3} + \left(8 a^{2} + 62 a + 25\right)\cdot 73^{4} + \left(29 a^{2} + 42 a + 36\right)\cdot 73^{5} + \left(28 a^{2} + 3 a + 54\right)\cdot 73^{6} + \left(58 a^{2} + 58 a + 60\right)\cdot 73^{7} + \left(60 a^{2} + 67 a + 13\right)\cdot 73^{8} + \left(48 a^{2} + 41 a + 69\right)\cdot 73^{9} +O(73^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,6)(3,5)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,5)(4,6)(7,9)$ | $2$ |
| $6$ | $3$ | $(3,8,5)$ | $0$ |
| $8$ | $3$ | $(1,6,9)(2,4,7)(3,5,8)$ | $3$ |
| $12$ | $3$ | $(1,9,6)(3,8,5)$ | $-3$ |
| $72$ | $3$ | $(1,3,2)(4,6,5)(7,9,8)$ | $0$ |
| $54$ | $4$ | $(1,3,6,5)(8,9)$ | $0$ |
| $54$ | $6$ | $(1,6)(2,4)(3,5,8)$ | $0$ |
| $108$ | $6$ | $(1,3,9,8,6,5)(2,4)$ | $-1$ |
| $72$ | $9$ | $(1,3,4,6,5,7,9,8,2)$ | $0$ |
| $72$ | $9$ | $(1,3,7,9,8,4,6,5,2)$ | $0$ |
| $54$ | $12$ | $(1,2,6,4)(3,8,5)(7,9)$ | $0$ |
| $54$ | $12$ | $(1,2,6,4)(3,5,8)(7,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.