Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(19871009045337088\)\(\medspace = 2^{10} \cdot 23^{4} \cdot 37^{5} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.3.11035059968.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | even |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.3.11035059968.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{3} + 2x + 68 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 36 a^{2} + 16 a + 40 + \left(58 a^{2} + 39 a + 33\right)\cdot 73 + \left(7 a^{2} + 20 a + 13\right)\cdot 73^{2} + \left(6 a^{2} + 39 a + 9\right)\cdot 73^{3} + \left(44 a^{2} + 21 a + 4\right)\cdot 73^{4} + \left(70 a^{2} + 31 a + 58\right)\cdot 73^{5} + \left(40 a^{2} + 34 a + 11\right)\cdot 73^{6} + \left(54 a^{2} + 24 a + 24\right)\cdot 73^{7} + \left(7 a^{2} + 38 a + 44\right)\cdot 73^{8} + \left(30 a^{2} + 58 a + 17\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 65 a^{2} + 15 a + 30 + \left(45 a^{2} + 46 a + 65\right)\cdot 73 + \left(22 a^{2} + 58 a + 8\right)\cdot 73^{2} + \left(8 a^{2} + 22 a + 12\right)\cdot 73^{3} + \left(51 a^{2} + 71 a + 62\right)\cdot 73^{4} + \left(39 a^{2} + 40 a + 16\right)\cdot 73^{5} + \left(35 a^{2} + 67 a + 53\right)\cdot 73^{6} + \left(42 a^{2} + 20 a + 56\right)\cdot 73^{7} + \left(45 a^{2} + 54 a + 21\right)\cdot 73^{8} + \left(21 a^{2} + 26 a + 6\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 27 a^{2} + 48 a + 59 + \left(34 a^{2} + 62 a\right)\cdot 73 + \left(66 a^{2} + 28 a + 8\right)\cdot 73^{2} + \left(17 a^{2} + 17 a + 72\right)\cdot 73^{3} + \left(4 a^{2} + 50 a + 66\right)\cdot 73^{4} + \left(66 a^{2} + 28 a + 16\right)\cdot 73^{5} + \left(7 a^{2} + 13 a + 37\right)\cdot 73^{6} + \left(60 a^{2} + 39 a + 16\right)\cdot 73^{7} + \left(34 a + 42\right)\cdot 73^{8} + \left(25 a^{2} + 23 a + 38\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 27 a^{2} + 22 + \left(36 a^{2} + 11 a + 16\right)\cdot 73 + \left(20 a^{2} + 15 a + 56\right)\cdot 73^{2} + \left(61 a^{2} + 53 a + 56\right)\cdot 73^{3} + \left(13 a^{2} + 37 a + 11\right)\cdot 73^{4} + \left(63 a^{2} + 51 a + 21\right)\cdot 73^{5} + \left(55 a^{2} + 38 a + 66\right)\cdot 73^{6} + \left(6 a^{2} + 13 a + 23\right)\cdot 73^{7} + \left(24 a^{2} + 37 a + 54\right)\cdot 73^{8} + \left(11 a^{2} + 36 a + 56\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 63 a^{2} + 34 + \left(60 a^{2} + 29 a + 60\right)\cdot 73 + \left(4 a^{2} + 56 a + 71\right)\cdot 73^{2} + \left(72 a^{2} + 51 a + 46\right)\cdot 73^{3} + \left(46 a^{2} + 7 a + 26\right)\cdot 73^{4} + \left(38 a^{2} + 72 a + 53\right)\cdot 73^{5} + \left(27 a^{2} + 61 a + 14\right)\cdot 73^{6} + \left(29 a^{2} + 10 a + 24\right)\cdot 73^{7} + \left(59 a^{2} + 37 a + 47\right)\cdot 73^{8} + \left(30 a^{2} + 37 a + 70\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a^{2} + 42 a + 18 + \left(62 a^{2} + 53 a + 2\right)\cdot 73 + \left(66 a^{2} + 35 a + 45\right)\cdot 73^{2} + \left(68 a^{2} + 2 a + 42\right)\cdot 73^{3} + \left(16 a^{2} + 57 a + 64\right)\cdot 73^{4} + \left(61 a^{2} + 70 a + 42\right)\cdot 73^{5} + \left(28 a^{2} + 68 a + 54\right)\cdot 73^{6} + \left(15 a^{2} + 23 a + 59\right)\cdot 73^{7} + \left(59 a^{2} + 49 a + 3\right)\cdot 73^{8} + \left(34 a^{2} + 71 a + 15\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 22 a^{2} + 31 a + 64 + \left(47 a^{2} + 8 a + 30\right)\cdot 73 + \left(58 a^{2} + 22 a + 58\right)\cdot 73^{2} + \left(15 a^{2} + 17 a + 44\right)\cdot 73^{3} + \left(42 a^{2} + 51 a + 49\right)\cdot 73^{4} + \left(21 a^{2} + 23 a + 38\right)\cdot 73^{5} + \left(61 a^{2} + 38 a\right)\cdot 73^{6} + \left(50 a^{2} + 35 a + 34\right)\cdot 73^{7} + \left(62 a^{2} + 59 a + 8\right)\cdot 73^{8} + \left(26 a^{2} + 37 a + 53\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 56 a^{2} + 25 a + 49 + \left(50 a^{2} + 54 a + 22\right)\cdot 73 + \left(a^{2} + 60 a + 43\right)\cdot 73^{2} + \left(56 a^{2} + 3 a + 25\right)\cdot 73^{3} + \left(21 a^{2} + 15 a + 17\right)\cdot 73^{4} + \left(41 a^{2} + 45 a + 8\right)\cdot 73^{5} + \left(37 a^{2} + 70 a + 28\right)\cdot 73^{6} + \left(56 a^{2} + 22 a + 60\right)\cdot 73^{7} + \left(12 a^{2} + a + 33\right)\cdot 73^{8} + \left(17 a^{2} + 12 a + 52\right)\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 45 a^{2} + 42 a + 52 + \left(41 a^{2} + 60 a + 59\right)\cdot 73 + \left(42 a^{2} + 66 a + 59\right)\cdot 73^{2} + \left(58 a^{2} + 10 a + 54\right)\cdot 73^{3} + \left(50 a^{2} + 53 a + 61\right)\cdot 73^{4} + \left(35 a^{2} + 35\right)\cdot 73^{5} + \left(69 a^{2} + 44 a + 25\right)\cdot 73^{6} + \left(48 a^{2} + 27 a + 65\right)\cdot 73^{7} + \left(19 a^{2} + 53 a + 35\right)\cdot 73^{8} + \left(21 a^{2} + 60 a + 54\right)\cdot 73^{9} +O(73^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(1,5)$ | $4$ |
| $18$ | $2$ | $(1,7)(5,8)(6,9)$ | $2$ |
| $27$ | $2$ | $(1,5)(2,3)(7,8)$ | $0$ |
| $27$ | $2$ | $(1,5)(2,3)$ | $0$ |
| $54$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $2$ |
| $6$ | $3$ | $(2,3,4)$ | $0$ |
| $8$ | $3$ | $(1,6,5)(2,4,3)(7,9,8)$ | $3$ |
| $12$ | $3$ | $(1,6,5)(2,4,3)$ | $-3$ |
| $72$ | $3$ | $(1,7,2)(3,5,8)(4,6,9)$ | $0$ |
| $54$ | $4$ | $(1,2,5,3)(4,6)$ | $0$ |
| $162$ | $4$ | $(1,2,5,3)(4,6)(7,8)$ | $0$ |
| $36$ | $6$ | $(1,7)(2,3,4)(5,8)(6,9)$ | $2$ |
| $36$ | $6$ | $(1,2,6,4,5,3)$ | $-1$ |
| $36$ | $6$ | $(1,5)(2,3,4)$ | $-2$ |
| $36$ | $6$ | $(1,5)(2,3,4)(7,8,9)$ | $1$ |
| $54$ | $6$ | $(1,5)(2,4,3)(7,8)$ | $0$ |
| $72$ | $6$ | $(1,7,6,9,5,8)(2,3,4)$ | $-1$ |
| $108$ | $6$ | $(1,2,6,4,5,3)(7,8)$ | $-1$ |
| $216$ | $6$ | $(1,7,2,5,8,3)(4,6,9)$ | $0$ |
| $144$ | $9$ | $(1,7,2,6,9,4,5,8,3)$ | $0$ |
| $108$ | $12$ | $(1,7,5,8)(2,3,4)(6,9)$ | $0$ |