Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(1442319106405376\)\(\medspace = 2^{10} \cdot 269^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.1340445266176.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | even |
| Determinant: | 1.269.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.1340445266176.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 4x^{7} - 2x^{6} + 2x^{5} - 24x^{4} - 8x^{3} + 80x^{2} + 68x + 4 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 20 + 69\cdot 73 + 4\cdot 73^{2} + 22\cdot 73^{3} + 11\cdot 73^{4} + 30\cdot 73^{5} + 43\cdot 73^{6} + 49\cdot 73^{7} + 22\cdot 73^{8} + 70\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 69 a^{2} + 59 a + 13 + \left(3 a^{2} + 53 a + 70\right)\cdot 73 + \left(52 a^{2} + 5 a + 22\right)\cdot 73^{2} + \left(28 a^{2} + 21 a + 29\right)\cdot 73^{3} + \left(9 a^{2} + 60 a + 71\right)\cdot 73^{4} + \left(70 a^{2} + 23 a + 52\right)\cdot 73^{5} + \left(19 a^{2} + 52 a + 5\right)\cdot 73^{6} + \left(38 a^{2} + 64 a + 63\right)\cdot 73^{7} + \left(3 a^{2} + 67 a + 34\right)\cdot 73^{8} + \left(42 a^{2} + 26 a + 7\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 24 a^{2} + 10 a + 26 + \left(a^{2} + 11 a + 42\right)\cdot 73 + \left(47 a^{2} + 24 a + 40\right)\cdot 73^{2} + \left(18 a^{2} + 18 a + 64\right)\cdot 73^{3} + \left(20 a^{2} + 42 a + 12\right)\cdot 73^{4} + \left(36 a^{2} + 69 a + 32\right)\cdot 73^{5} + \left(17 a^{2} + 58 a + 2\right)\cdot 73^{6} + \left(22 a^{2} + 58 a + 66\right)\cdot 73^{7} + \left(9 a^{2} + 58 a + 66\right)\cdot 73^{8} + \left(24 a^{2} + 7 a + 7\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 53 a^{2} + 4 a + 16 + \left(67 a^{2} + 8 a + 9\right)\cdot 73 + \left(46 a^{2} + 43 a + 16\right)\cdot 73^{2} + \left(25 a^{2} + 33 a + 25\right)\cdot 73^{3} + \left(43 a^{2} + 43 a + 19\right)\cdot 73^{4} + \left(39 a^{2} + 52 a + 12\right)\cdot 73^{5} + \left(35 a^{2} + 34 a + 2\right)\cdot 73^{6} + \left(12 a^{2} + 22 a + 53\right)\cdot 73^{7} + \left(60 a^{2} + 19 a + 61\right)\cdot 73^{8} + \left(6 a^{2} + 38 a + 57\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 68 + 14\cdot 73 + 62\cdot 73^{2} + 5\cdot 73^{3} + 7\cdot 73^{4} + 58\cdot 73^{5} + 56\cdot 73^{6} + 24\cdot 73^{7} + 44\cdot 73^{8} + 2\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 a^{2} + 47 a + 27 + \left(40 a^{2} + 21 a + 19\right)\cdot 73 + \left(51 a^{2} + 53 a + 68\right)\cdot 73^{2} + \left(39 a^{2} + 9 a + 3\right)\cdot 73^{3} + \left(57 a^{2} + 10 a + 4\right)\cdot 73^{4} + \left(48 a^{2} + 53 a + 70\right)\cdot 73^{5} + \left(9 a^{2} + 64 a + 17\right)\cdot 73^{6} + \left(12 a^{2} + 3 a + 38\right)\cdot 73^{7} + \left(7 a^{2} + 29 a + 27\right)\cdot 73^{8} + \left(2 a^{2} + 67 a + 65\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 + 45\cdot 73 + 23\cdot 73^{4} + 18\cdot 73^{5} + 20\cdot 73^{6} + 42\cdot 73^{7} + 7\cdot 73^{8} + 31\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 53 a^{2} + 27 a + 61 + \left(68 a^{2} + 35 a + 7\right)\cdot 73 + \left(59 a^{2} + 43 a + 55\right)\cdot 73^{2} + \left(2 a^{2} + 57 a + 27\right)\cdot 73^{3} + \left(13 a^{2} + 49 a + 66\right)\cdot 73^{4} + \left(62 a^{2} + 24 a + 14\right)\cdot 73^{5} + \left(30 a^{2} + 57 a + 46\right)\cdot 73^{6} + \left(a^{2} + 16 a + 72\right)\cdot 73^{7} + \left(a^{2} + 48 a + 67\right)\cdot 73^{8} + \left(a^{2} + 63\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 29 a^{2} + 72 a + 29 + \left(36 a^{2} + 15 a + 13\right)\cdot 73 + \left(34 a^{2} + 49 a + 21\right)\cdot 73^{2} + \left(30 a^{2} + 5 a + 40\right)\cdot 73^{3} + \left(2 a^{2} + 13 a + 3\right)\cdot 73^{4} + \left(35 a^{2} + 68 a + 3\right)\cdot 73^{5} + \left(32 a^{2} + 23 a + 24\right)\cdot 73^{6} + \left(59 a^{2} + 52 a + 28\right)\cdot 73^{7} + \left(64 a^{2} + 68 a + 31\right)\cdot 73^{8} + \left(69 a^{2} + 4 a + 58\right)\cdot 73^{9} +O(73^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,5)(6,8)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,5)(4,7)(6,8)$ | $2$ |
| $6$ | $3$ | $(6,9,8)$ | $0$ |
| $8$ | $3$ | $(1,5,7)(2,3,4)(6,8,9)$ | $3$ |
| $12$ | $3$ | $(1,7,5)(6,9,8)$ | $-3$ |
| $72$ | $3$ | $(1,6,2)(3,5,8)(4,7,9)$ | $0$ |
| $54$ | $4$ | $(1,6,5,8)(7,9)$ | $0$ |
| $54$ | $6$ | $(1,5)(2,3)(6,8,9)$ | $0$ |
| $108$ | $6$ | $(1,6,7,9,5,8)(2,3)$ | $-1$ |
| $72$ | $9$ | $(1,6,3,5,8,4,7,9,2)$ | $0$ |
| $72$ | $9$ | $(1,6,4,7,9,3,5,8,2)$ | $0$ |
| $54$ | $12$ | $(1,2,5,3)(4,7)(6,9,8)$ | $0$ |
| $54$ | $12$ | $(1,2,5,3)(4,7)(6,8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.