Basic invariants
| Dimension: | $12$ |
| Group: | $S_3 \wr C_3 $ |
| Conductor: | \(88605052913988081\)\(\medspace = 3^{20} \cdot 71^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.3056317191.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T206 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr C_3$ |
| Projective stem field: | Galois closure of 9.3.3056317191.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 6x^{7} - 9x^{6} + 6x^{5} - 3x^{4} - 3x^{3} + 6x^{2} - 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{3} + 9x + 76 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 a^{2} + 73 a + 4 + \left(47 a^{2} + 20 a + 50\right)\cdot 79 + \left(74 a^{2} + 29 a + 78\right)\cdot 79^{2} + \left(33 a^{2} + 17 a + 71\right)\cdot 79^{3} + \left(16 a^{2} + 54 a + 31\right)\cdot 79^{4} + \left(20 a^{2} + 40 a + 64\right)\cdot 79^{5} + \left(74 a^{2} + 50 a + 65\right)\cdot 79^{6} + \left(72 a^{2} + 23 a + 17\right)\cdot 79^{7} + \left(10 a^{2} + 21 a + 42\right)\cdot 79^{8} + \left(64 a^{2} + 29 a + 73\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 44 a^{2} + 54 a + 76 + \left(77 a^{2} + 27 a + 77\right)\cdot 79 + \left(4 a^{2} + 34 a + 17\right)\cdot 79^{2} + \left(2 a^{2} + 30 a + 2\right)\cdot 79^{3} + \left(60 a^{2} + 50 a + 31\right)\cdot 79^{4} + \left(17 a^{2} + 26 a + 62\right)\cdot 79^{5} + \left(61 a^{2} + 73 a + 54\right)\cdot 79^{6} + \left(55 a^{2} + 9 a + 41\right)\cdot 79^{7} + \left(2 a^{2} + 54 a + 9\right)\cdot 79^{8} + \left(2 a^{2} + 17 a + 67\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 52 a^{2} + 45 + \left(6 a^{2} + 12 a + 47\right)\cdot 79 + \left(46 a^{2} + 56 a + 27\right)\cdot 79^{2} + \left(55 a^{2} + 16 a + 7\right)\cdot 79^{3} + \left(46 a^{2} + 5 a + 30\right)\cdot 79^{4} + \left(28 a^{2} + 22 a + 48\right)\cdot 79^{5} + \left(12 a^{2} + 74 a + 77\right)\cdot 79^{6} + \left(29 a^{2} + 56 a + 39\right)\cdot 79^{7} + \left(28 a^{2} + 13 a + 5\right)\cdot 79^{8} + \left(21 a^{2} + 67 a + 25\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 32 a^{2} + 74 a + 41 + \left(54 a^{2} + 62 a + 78\right)\cdot 79 + \left(9 a^{2} + 52 a + 43\right)\cdot 79^{2} + \left(7 a^{2} + 73 a + 26\right)\cdot 79^{3} + \left(17 a^{2} + 28 a + 24\right)\cdot 79^{4} + \left(48 a^{2} + 55 a + 74\right)\cdot 79^{5} + \left(23 a^{2} + 39 a + 43\right)\cdot 79^{6} + \left(74 a^{2} + 52\right)\cdot 79^{7} + \left(64 a^{2} + 44 a + 24\right)\cdot 79^{8} + \left(19 a^{2} + 26 a + 59\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 66 a^{2} + 38 a + 25 + \left(37 a^{2} + 34 a + 72\right)\cdot 79 + \left(41 a^{2} + 5 a + 37\right)\cdot 79^{2} + \left(40 a^{2} + 76 a + 32\right)\cdot 79^{3} + \left(41 a^{2} + 19 a + 24\right)\cdot 79^{4} + \left(23 a^{2} + a + 5\right)\cdot 79^{5} + \left(36 a^{2} + 31 a + 75\right)\cdot 79^{6} + \left(71 a^{2} + 39 a + 8\right)\cdot 79^{7} + \left(48 a^{2} + 28 a + 33\right)\cdot 79^{8} + \left(61 a^{2} + 44 a + 58\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 62 a^{2} + 25 a + 26 + \left(73 a^{2} + 39 a + 55\right)\cdot 79 + \left(27 a^{2} + 67 a + 76\right)\cdot 79^{2} + \left(21 a^{2} + 31 a + 38\right)\cdot 79^{3} + \left(51 a^{2} + 23 a + 57\right)\cdot 79^{4} + \left(32 a^{2} + 30 a + 72\right)\cdot 79^{5} + \left(5 a^{2} + 10 a + 35\right)\cdot 79^{6} + \left(73 a^{2} + 12 a + 66\right)\cdot 79^{7} + \left(47 a^{2} + 11 a + 43\right)\cdot 79^{8} + \left(55 a^{2} + 73 a + 72\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 49 a^{2} + 78 a + 64 + \left(39 a^{2} + 9 a + 68\right)\cdot 79 + \left(57 a^{2} + 14\right)\cdot 79^{2} + \left(50 a^{2} + 69 a + 51\right)\cdot 79^{3} + \left(13 a^{2} + 57 a + 3\right)\cdot 79^{4} + \left(77 a^{2} + 54 a + 11\right)\cdot 79^{5} + \left(78 a^{2} + 40 a + 60\right)\cdot 79^{6} + \left(38 a^{2} + 24 a + 77\right)\cdot 79^{7} + \left(28 a^{2} + 66 a + 42\right)\cdot 79^{8} + \left(12 a^{2} + 23 a + 14\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 69 a^{2} + 47 a + 43 + \left(72 a^{2} + 23 a + 45\right)\cdot 79 + \left(41 a^{2} + 44 a + 40\right)\cdot 79^{2} + \left(4 a^{2} + 64 a + 53\right)\cdot 79^{3} + \left(21 a^{2} + 4 a + 59\right)\cdot 79^{4} + \left(35 a^{2} + 37 a + 75\right)\cdot 79^{5} + \left(47 a^{2} + 76 a + 62\right)\cdot 79^{6} + \left(13 a^{2} + 15 a + 56\right)\cdot 79^{7} + \left(19 a^{2} + 29 a + 12\right)\cdot 79^{8} + \left(32 a^{2} + 5 a + 40\right)\cdot 79^{9} +O(79^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 77 a^{2} + 6 a + 74 + \left(63 a^{2} + 6 a + 56\right)\cdot 79 + \left(11 a^{2} + 26 a + 56\right)\cdot 79^{2} + \left(21 a^{2} + 15 a + 31\right)\cdot 79^{3} + \left(48 a^{2} + 71 a + 53\right)\cdot 79^{4} + \left(32 a^{2} + 47 a + 59\right)\cdot 79^{5} + \left(55 a^{2} + 77 a + 76\right)\cdot 79^{6} + \left(44 a^{2} + 53 a + 32\right)\cdot 79^{7} + \left(64 a^{2} + 47 a + 22\right)\cdot 79^{8} + \left(46 a^{2} + 28 a + 63\right)\cdot 79^{9} +O(79^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(3,7)$ | $4$ |
| $27$ | $2$ | $(1,2)(3,7)(5,6)$ | $0$ |
| $27$ | $2$ | $(2,4)(3,7)$ | $0$ |
| $6$ | $3$ | $(5,6,9)$ | $0$ |
| $8$ | $3$ | $(1,4,2)(3,8,7)(5,9,6)$ | $3$ |
| $12$ | $3$ | $(1,2,4)(5,6,9)$ | $-3$ |
| $36$ | $3$ | $(1,5,3)(2,6,7)(4,9,8)$ | $0$ |
| $36$ | $3$ | $(1,3,5)(2,7,6)(4,8,9)$ | $0$ |
| $18$ | $6$ | $(3,7)(5,6,9)$ | $-2$ |
| $18$ | $6$ | $(1,2,4)(3,7)$ | $-2$ |
| $36$ | $6$ | $(1,2,4)(3,7)(5,6,9)$ | $1$ |
| $54$ | $6$ | $(2,4)(3,7)(5,6,9)$ | $0$ |
| $108$ | $6$ | $(1,5,7,2,6,3)(4,9,8)$ | $0$ |
| $108$ | $6$ | $(1,3,6,2,7,5)(4,8,9)$ | $0$ |
| $72$ | $9$ | $(1,5,8,4,9,7,2,6,3)$ | $0$ |
| $72$ | $9$ | $(1,8,9,2,3,5,4,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.