Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(56773591412619443\)\(\medspace = 2243^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.25311454040401.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | odd |
| Determinant: | 1.2243.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.1.25311454040401.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 6x^{7} - 18x^{6} + 41x^{5} - 51x^{4} + 106x^{3} - 116x^{2} + 72x + 16 \)
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The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{3} + 9x + 92 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( a + 17 + \left(12 a^{2} + 37 a + 81\right)\cdot 97 + \left(92 a^{2} + 21 a + 64\right)\cdot 97^{2} + \left(91 a^{2} + 50 a + 10\right)\cdot 97^{3} + \left(85 a^{2} + 67 a + 55\right)\cdot 97^{4} + \left(26 a^{2} + 93 a + 9\right)\cdot 97^{5} + \left(65 a^{2} + 96 a + 66\right)\cdot 97^{6} + \left(84 a^{2} + 45 a + 78\right)\cdot 97^{7} + \left(71 a^{2} + 45 a + 91\right)\cdot 97^{8} + \left(76 a^{2} + 38 a + 79\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 13 a^{2} + 75 a + 95 + \left(91 a^{2} + 70 a + 70\right)\cdot 97 + \left(18 a^{2} + 61 a + 13\right)\cdot 97^{2} + \left(20 a^{2} + 31 a + 65\right)\cdot 97^{3} + \left(58 a^{2} + 61 a + 82\right)\cdot 97^{4} + \left(65 a^{2} + 46 a + 47\right)\cdot 97^{5} + \left(57 a^{2} + 24 a + 20\right)\cdot 97^{6} + \left(28 a^{2} + 35 a + 33\right)\cdot 97^{7} + \left(a^{2} + 27 a + 56\right)\cdot 97^{8} + \left(14 a^{2} + 81 a + 91\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 29 + 52\cdot 97 + 14\cdot 97^{2} + 7\cdot 97^{3} + 92\cdot 97^{4} + 91\cdot 97^{5} + 38\cdot 97^{6} + 80\cdot 97^{7} + 24\cdot 97^{8} + 20\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 88 a^{2} + 48 a + 52 + \left(79 a^{2} + 3 a + 38\right)\cdot 97 + \left(43 a^{2} + 78 a + 91\right)\cdot 97^{2} + \left(31 a^{2} + 18 a + 71\right)\cdot 97^{3} + \left(87 a^{2} + 2 a + 13\right)\cdot 97^{4} + \left(14 a^{2} + 27 a + 46\right)\cdot 97^{5} + \left(5 a^{2} + 71 a + 79\right)\cdot 97^{6} + \left(56 a^{2} + 45 a + 32\right)\cdot 97^{7} + \left(37 a^{2} + 46 a + 84\right)\cdot 97^{8} + \left(56 a^{2} + 81 a + 14\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 57 + 49\cdot 97 + 31\cdot 97^{2} + 91\cdot 97^{3} + 18\cdot 97^{4} + 97^{5} + 25\cdot 97^{6} + 95\cdot 97^{7} + 80\cdot 97^{8} + 70\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 47 a^{2} + 48 a + \left(74 a^{2} + 37 a + 6\right)\cdot 97 + \left(29 a^{2} + 46 a + 7\right)\cdot 97^{2} + \left(85 a^{2} + 46 a + 7\right)\cdot 97^{3} + \left(62 a^{2} + 14 a + 61\right)\cdot 97^{4} + \left(80 a^{2} + 93 a + 52\right)\cdot 97^{5} + \left(6 a^{2} + 36 a + 89\right)\cdot 97^{6} + \left(74 a^{2} + 73 a + 43\right)\cdot 97^{7} + \left(41 a^{2} + 11 a + 12\right)\cdot 97^{8} + \left(92 a^{2} + 74 a + 37\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 + 29\cdot 97 + 89\cdot 97^{2} + 31\cdot 97^{3} + 85\cdot 97^{4} + 6\cdot 97^{5} + 88\cdot 97^{6} + 82\cdot 97^{7} + 74\cdot 97^{8} + 80\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 84 a^{2} + 21 a + 36 + \left(90 a^{2} + 86 a + 69\right)\cdot 97 + \left(82 a^{2} + 13 a + 9\right)\cdot 97^{2} + \left(81 a^{2} + 15 a + 47\right)\cdot 97^{3} + \left(49 a^{2} + 65 a + 32\right)\cdot 97^{4} + \left(4 a^{2} + 53 a + 69\right)\cdot 97^{5} + \left(71 a^{2} + 72 a + 3\right)\cdot 97^{6} + \left(80 a^{2} + 15 a + 55\right)\cdot 97^{7} + \left(23 a^{2} + 24 a + 94\right)\cdot 97^{8} + \left(6 a^{2} + 74 a + 44\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 59 a^{2} + a + 72 + \left(39 a^{2} + 56 a + 87\right)\cdot 97 + \left(23 a^{2} + 69 a + 65\right)\cdot 97^{2} + \left(77 a^{2} + 31 a + 55\right)\cdot 97^{3} + \left(43 a^{2} + 80 a + 43\right)\cdot 97^{4} + \left(a^{2} + 73 a + 62\right)\cdot 97^{5} + \left(85 a^{2} + 85 a + 73\right)\cdot 97^{6} + \left(63 a^{2} + 74 a + 79\right)\cdot 97^{7} + \left(17 a^{2} + 38 a + 61\right)\cdot 97^{8} + \left(45 a^{2} + 38 a + 44\right)\cdot 97^{9} +O(97^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(3,5)(4,6)$ | $0$ |
| $54$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $2$ |
| $6$ | $3$ | $(4,9,6)$ | $0$ |
| $8$ | $3$ | $(1,2,8)(3,5,7)(4,6,9)$ | $3$ |
| $12$ | $3$ | $(3,7,5)(4,9,6)$ | $-3$ |
| $72$ | $3$ | $(1,3,4)(2,5,6)(7,9,8)$ | $0$ |
| $54$ | $4$ | $(3,4,5,6)(7,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(3,5)(4,6,9)$ | $0$ |
| $108$ | $6$ | $(1,2)(3,4,7,9,5,6)$ | $-1$ |
| $72$ | $9$ | $(1,3,4,2,5,6,8,7,9)$ | $0$ |
| $72$ | $9$ | $(1,3,4,8,7,9,2,5,6)$ | $0$ |
| $54$ | $12$ | $(1,5,2,3)(4,9,6)(7,8)$ | $0$ |
| $54$ | $12$ | $(1,5,2,3)(4,6,9)(7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.