Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(9121115669451776\)\(\medspace = 2^{10} \cdot 389^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.5861899530496.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | even |
| Determinant: | 1.389.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.5.5861899530496.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} - 2x^{7} + 12x^{6} - 12x^{5} - 6x^{4} + 16x^{3} - 4x^{2} - 5x + 1 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 57 a^{2} + 31 a + 61 + \left(67 a^{2} + 66 a + 23\right)\cdot 73 + \left(47 a^{2} + 21 a + 26\right)\cdot 73^{2} + \left(50 a^{2} + 42 a + 4\right)\cdot 73^{3} + \left(22 a^{2} + 50 a + 9\right)\cdot 73^{4} + \left(67 a^{2} + 19 a + 64\right)\cdot 73^{5} + \left(10 a^{2} + 48 a + 13\right)\cdot 73^{6} + \left(46 a^{2} + 20 a + 71\right)\cdot 73^{7} + \left(50 a^{2} + 49 a + 29\right)\cdot 73^{8} + \left(55 a^{2} + 57 a + 11\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 37 a^{2} + 12 a + 46 + \left(29 a^{2} + 47 a + 40\right)\cdot 73 + \left(62 a^{2} + 58 a + 31\right)\cdot 73^{2} + \left(26 a^{2} + 32 a + 6\right)\cdot 73^{3} + \left(45 a^{2} + 40 a + 44\right)\cdot 73^{4} + \left(36 a^{2} + 69 a + 34\right)\cdot 73^{5} + \left(35 a^{2} + 52 a + 51\right)\cdot 73^{6} + \left(47 a^{2} + 50 a + 68\right)\cdot 73^{7} + \left(4 a^{2} + 43 a + 18\right)\cdot 73^{8} + \left(64 a^{2} + 41 a + 50\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 50 a^{2} + 22 a + 3 + \left(20 a^{2} + 69 a + 34\right)\cdot 73 + \left(58 a^{2} + 24 a + 64\right)\cdot 73^{2} + \left(33 a^{2} + 11 a + 54\right)\cdot 73^{3} + \left(46 a^{2} + 25 a + 40\right)\cdot 73^{4} + \left(57 a^{2} + 40 a + 2\right)\cdot 73^{5} + \left(49 a^{2} + 38 a + 17\right)\cdot 73^{6} + \left(52 a^{2} + 31\right)\cdot 73^{7} + \left(25 a^{2} + 3 a + 45\right)\cdot 73^{8} + \left(38 a^{2} + 4 a + 12\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a^{2} + 20 a + 37 + \left(57 a^{2} + 10 a + 34\right)\cdot 73 + \left(39 a^{2} + 26 a + 15\right)\cdot 73^{2} + \left(61 a^{2} + 19 a + 43\right)\cdot 73^{3} + \left(3 a^{2} + 70 a + 32\right)\cdot 73^{4} + \left(21 a^{2} + 12 a + 2\right)\cdot 73^{5} + \left(12 a^{2} + 59 a + 40\right)\cdot 73^{6} + \left(47 a^{2} + 51 a + 72\right)\cdot 73^{7} + \left(69 a^{2} + 20 a + 30\right)\cdot 73^{8} + \left(51 a^{2} + 11 a + 6\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 a^{2} + 18 a + 50 + \left(52 a^{2} + 16 a + 22\right)\cdot 73 + \left(24 a^{2} + 52 a + 54\right)\cdot 73^{2} + \left(36 a^{2} + 15 a + 67\right)\cdot 73^{3} + \left(14 a^{2} + 7 a + 2\right)\cdot 73^{4} + \left(26 a^{2} + 54 a + 45\right)\cdot 73^{5} + \left(15 a^{2} + 16 a + 24\right)\cdot 73^{6} + \left(46 a^{2} + 29 a + 18\right)\cdot 73^{7} + \left(44 a^{2} + 52 a + 72\right)\cdot 73^{8} + \left(17 a^{2} + 49 a + 36\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 56 a^{2} + 33 a + 21 + \left(72 a^{2} + 60 a + 16\right)\cdot 73 + \left(62 a^{2} + 68 a + 27\right)\cdot 73^{2} + \left(2 a^{2} + 45 a + 23\right)\cdot 73^{3} + \left(12 a^{2} + 40 a + 29\right)\cdot 73^{4} + \left(62 a^{2} + 51 a + 25\right)\cdot 73^{5} + \left(7 a^{2} + 17 a + 31\right)\cdot 73^{6} + \left(47 a^{2} + 43 a + 23\right)\cdot 73^{7} + \left(2 a^{2} + 17 a + 28\right)\cdot 73^{8} + \left(17 a^{2} + 19 a + 23\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 20 a^{2} + 72 a + 46 + \left(14 a^{2} + 69 a + 35\right)\cdot 73 + \left(39 a^{2} + 15 a + 68\right)\cdot 73^{2} + \left(12 a^{2} + 6 a + 11\right)\cdot 73^{3} + \left(37 a^{2} + 70 a + 14\right)\cdot 73^{4} + \left(68 a^{2} + 30 a + 58\right)\cdot 73^{5} + \left(39 a^{2} + 21 a + 49\right)\cdot 73^{6} + \left(47 a^{2} + 59 a + 72\right)\cdot 73^{7} + \left(71 a^{2} + 46 a + 22\right)\cdot 73^{8} + \left(25 a^{2} + 33 a + 35\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 69 a^{2} + 43 a + 40 + \left(63 a^{2} + 9 a + 13\right)\cdot 73 + \left(58 a^{2} + 35 a + 51\right)\cdot 73^{2} + \left(9 a^{2} + 24 a + 56\right)\cdot 73^{3} + \left(13 a^{2} + 25 a + 49\right)\cdot 73^{4} + \left(10 a^{2} + 22 a + 23\right)\cdot 73^{5} + \left(22 a^{2} + 3 a + 9\right)\cdot 73^{6} + \left(52 a^{2} + 66 a + 2\right)\cdot 73^{7} + \left(23 a^{2} + 49 a + 20\right)\cdot 73^{8} + \left(64 a^{2} + 54 a + 26\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 70 a^{2} + 41 a + 64 + \left(58 a^{2} + 15 a + 70\right)\cdot 73 + \left(43 a^{2} + 61 a + 25\right)\cdot 73^{2} + \left(57 a^{2} + 20 a + 23\right)\cdot 73^{3} + \left(23 a^{2} + 35 a + 69\right)\cdot 73^{4} + \left(15 a^{2} + 63 a + 35\right)\cdot 73^{5} + \left(25 a^{2} + 33 a + 54\right)\cdot 73^{6} + \left(51 a^{2} + 43 a + 4\right)\cdot 73^{7} + \left(71 a^{2} + 8 a + 23\right)\cdot 73^{8} + \left(29 a^{2} + 20 a + 16\right)\cdot 73^{9} +O(73^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,7)(3,5)$ | $0$ |
| $54$ | $2$ | $(1,7)(2,3)(4,5)(6,9)$ | $2$ |
| $6$ | $3$ | $(1,8,7)$ | $0$ |
| $8$ | $3$ | $(1,7,8)(2,4,9)(3,5,6)$ | $3$ |
| $12$ | $3$ | $(1,8,7)(3,6,5)$ | $-3$ |
| $72$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $0$ |
| $54$ | $4$ | $(1,5,7,3)(6,8)$ | $0$ |
| $54$ | $6$ | $(1,7,8)(2,4)(3,5)$ | $0$ |
| $108$ | $6$ | $(1,6,8,5,7,3)(2,4)$ | $-1$ |
| $72$ | $9$ | $(1,4,5,7,9,6,8,2,3)$ | $0$ |
| $72$ | $9$ | $(1,9,6,8,4,5,7,2,3)$ | $0$ |
| $54$ | $12$ | $(1,8,7)(2,5,4,3)(6,9)$ | $0$ |
| $54$ | $12$ | $(1,7,8)(2,5,4,3)(6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.