Basic invariants
Dimension: | $2$ |
Group: | $S_3 \times C_5$ |
Conductor: | \(116281\)\(\medspace = 11^{2} \cdot 31^{2} \) |
Artin stem field: | Galois closure of 15.5.28295791416461461425745668731.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | $S_3 \times C_5$ |
Parity: | odd |
Determinant: | 1.11.10t1.a.d |
Projective image: | $S_3$ |
Projective stem field: | Galois closure of 3.1.10571.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{15} - 5 x^{14} - 23 x^{13} - 164 x^{12} + 654 x^{11} + 2155 x^{10} + 7876 x^{9} - 2893 x^{8} + \cdots - 26797639 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 103 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 103 }$: \( x^{5} + 11x + 98 \)
Roots:
$r_{ 1 }$ | $=$ | \( 4 a^{4} + 45 a^{3} + 3 a^{2} + 53 a + 99 + \left(76 a^{4} + 102 a^{3} + 78 a^{2} + 96 a + 19\right)\cdot 103 + \left(75 a^{4} + 80 a^{3} + 65 a^{2} + 87 a + 101\right)\cdot 103^{2} + \left(26 a^{4} + 16 a^{3} + 102 a^{2} + 8 a + 22\right)\cdot 103^{3} + \left(19 a^{4} + 51 a^{3} + 11 a^{2} + 85 a + 61\right)\cdot 103^{4} + \left(7 a^{4} + 86 a^{3} + 58 a^{2} + 77 a + 87\right)\cdot 103^{5} + \left(98 a^{4} + 20 a^{3} + 80 a^{2} + 90 a + 65\right)\cdot 103^{6} + \left(11 a^{4} + 78 a^{3} + 73 a^{2} + 18 a + 102\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 2 }$ | $=$ | \( 7 a^{4} + 41 a^{3} + 40 a^{2} + 45 a + 75 + \left(95 a^{4} + 88 a^{3} + 26 a^{2} + 82 a + 17\right)\cdot 103 + \left(17 a^{4} + 85 a^{3} + 48 a^{2} + 74 a + 64\right)\cdot 103^{2} + \left(2 a^{4} + 14 a^{3} + 8 a^{2} + 9 a + 21\right)\cdot 103^{3} + \left(12 a^{4} + 49 a^{3} + 88 a^{2} + 3 a + 46\right)\cdot 103^{4} + \left(14 a^{4} + 34 a^{3} + 68 a^{2} + 31 a + 75\right)\cdot 103^{5} + \left(27 a^{4} + 91 a^{3} + 14 a^{2} + 92 a + 64\right)\cdot 103^{6} + \left(14 a^{4} + 91 a^{3} + 8 a^{2} + 86 a + 44\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 3 }$ | $=$ | \( 32 a^{4} + 100 a^{3} + 89 a^{2} + 52 a + 20 + \left(21 a^{4} + 64 a^{3} + 93 a^{2} + 76 a + 28\right)\cdot 103 + \left(66 a^{4} + 90 a^{3} + 31 a^{2} + 50 a + 67\right)\cdot 103^{2} + \left(36 a^{4} + 51 a^{3} + 23 a^{2} + 26 a + 99\right)\cdot 103^{3} + \left(61 a^{4} + 71 a^{3} + 95 a^{2} + 98 a + 27\right)\cdot 103^{4} + \left(32 a^{4} + 66 a^{3} + 29 a^{2} + 48 a + 2\right)\cdot 103^{5} + \left(92 a^{4} + 35 a^{3} + 24 a^{2} + 100 a + 12\right)\cdot 103^{6} + \left(101 a^{4} + 29 a^{3} + 31 a^{2} + 3 a + 53\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 4 }$ | $=$ | \( 40 a^{4} + 54 a^{3} + 54 a^{2} + 26 a + 77 + \left(29 a^{4} + 74 a^{3} + 75 a^{2} + 65 a + 57\right)\cdot 103 + \left(47 a^{4} + 12 a^{3} + 64 a^{2} + 9 a + 13\right)\cdot 103^{2} + \left(19 a^{4} + 75 a^{3} + 56 a^{2} + 9 a + 50\right)\cdot 103^{3} + \left(17 a^{4} + 58 a^{3} + 40 a^{2} + 64 a + 50\right)\cdot 103^{4} + \left(25 a^{4} + 84 a^{3} + 38 a^{2} + 91 a + 28\right)\cdot 103^{5} + \left(3 a^{4} + 46 a^{3} + 88 a^{2} + 77 a + 19\right)\cdot 103^{6} + \left(29 a^{4} + 16 a^{3} + 60 a^{2} + 16 a + 92\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 5 }$ | $=$ | \( 49 a^{4} + 5 a^{3} + 77 a^{2} + 81 a + 46 + \left(73 a^{4} + 40 a^{3} + 16 a^{2} + 61 a + 13\right)\cdot 103 + \left(85 a^{4} + 85 a^{3} + 96 a^{2} + 28 a + 74\right)\cdot 103^{2} + \left(91 a^{4} + 97 a^{3} + 89 a^{2} + 26 a + 49\right)\cdot 103^{3} + \left(46 a^{4} + 33 a^{3} + 38 a^{2} + 63 a + 65\right)\cdot 103^{4} + \left(29 a^{4} + 23 a^{3} + 21 a^{2} + 82 a + 77\right)\cdot 103^{5} + \left(86 a^{4} + 43 a^{3} + 75 a^{2} + 98 a + 20\right)\cdot 103^{6} + \left(44 a^{4} + 51 a^{3} + 63 a^{2} + 36 a + 4\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 6 }$ | $=$ | \( 61 a^{4} + 58 a^{3} + 49 a^{2} + 86 a + 65 + \left(35 a^{4} + 59 a^{3} + 97 a^{2} + 57 a + 96\right)\cdot 103 + \left(44 a^{4} + 94 a^{3} + 90 a^{2} + 86 a + 30\right)\cdot 103^{2} + \left(40 a^{4} + 2 a^{3} + 81 a^{2} + 69 a + 102\right)\cdot 103^{3} + \left(57 a^{4} + 59 a^{3} + 18 a^{2} + 11 a + 25\right)\cdot 103^{4} + \left(25 a^{4} + 83 a^{3} + 83 a^{2} + 21 a + 43\right)\cdot 103^{5} + \left(42 a^{4} + 40 a^{3} + 102 a^{2} + 85 a + 48\right)\cdot 103^{6} + \left(3 a^{4} + 68 a^{3} + 61 a^{2} + 42 a + 27\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 7 }$ | $=$ | \( 63 a^{4} + 29 a^{3} + 19 a^{2} + 89 a + 25 + \left(54 a^{4} + 51 a^{3} + 75 a^{2} + 82 a + 12\right)\cdot 103 + \left(97 a^{4} + 25 a^{3} + 26 a^{2} + 22 a + 75\right)\cdot 103^{2} + \left(44 a^{4} + 96 a^{3} + 86 a^{2} + 73 a + 69\right)\cdot 103^{3} + \left(87 a^{4} + 45 a^{3} + 51 a^{2} + 15 a + 92\right)\cdot 103^{4} + \left(6 a^{4} + 102 a^{3} + 55 a^{2} + 69 a + 22\right)\cdot 103^{5} + \left(72 a^{4} + 78 a^{3} + 59 a^{2} + 2 a + 19\right)\cdot 103^{6} + \left(51 a^{4} + 63 a^{3} + 40 a^{2} + 97 a + 85\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 8 }$ | $=$ | \( 66 a^{4} + 62 a^{3} + 40 a^{2} + 41 a + 6 + \left(83 a^{4} + 41 a^{3} + 26 a^{2} + 56 a + 25\right)\cdot 103 + \left(65 a^{4} + 20 a^{3} + 61 a^{2} + 59 a + 96\right)\cdot 103^{2} + \left(16 a^{4} + 85 a^{3} + 40 a^{2} + 34 a + 98\right)\cdot 103^{3} + \left(79 a^{4} + 75 a^{3} + 28 a^{2} + 32 a + 93\right)\cdot 103^{4} + \left(79 a^{4} + 39 a^{3} + 99 a^{2} + 23 a + 25\right)\cdot 103^{5} + \left(73 a^{4} + 45 a^{3} + 30 a^{2} + 48 a + 99\right)\cdot 103^{6} + \left(78 a^{4} + 50 a^{3} + 84 a^{2} + 88 a + 30\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 9 }$ | $=$ | \( 76 a^{4} + 60 a^{3} + 10 a^{2} + 100 a + 94 + \left(49 a^{4} + 19 a^{3} + 51 a^{2} + 75 a + 76\right)\cdot 103 + \left(69 a^{4} + 26 a^{3} + 99 a^{2} + 37 a + 66\right)\cdot 103^{2} + \left(10 a^{4} + 31 a^{3} + 72 a^{2} + 49 a + 87\right)\cdot 103^{3} + \left(66 a^{4} + 81 a^{3} + 60 a^{2} + 56 a + 40\right)\cdot 103^{4} + \left(79 a^{4} + 6 a^{3} + 51 a^{2} + 15 a + 45\right)\cdot 103^{5} + \left(94 a^{4} + 25 a^{3} + 84 a^{2} + 5 a + 57\right)\cdot 103^{6} + \left(27 a^{4} + 73 a^{3} + 76 a^{2} + 8 a + 16\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 10 }$ | $=$ | \( 78 a^{4} + 12 a^{3} + 83 a^{2} + 75 a + 20 + \left(34 a^{4} + 63 a^{3} + 100 a^{2} + 11 a + 43\right)\cdot 103 + \left(4 a^{4} + 16 a^{3} + 19 a^{2} + 17 a + 6\right)\cdot 103^{2} + \left(26 a^{4} + 93 a^{3} + 70 a^{2} + 63 a + 5\right)\cdot 103^{3} + \left(79 a^{4} + 30 a^{3} + 96 a^{2} + 67 a + 61\right)\cdot 103^{4} + \left(20 a^{4} + 55 a^{3} + 26 a^{2} + 53 a + 51\right)\cdot 103^{5} + \left(77 a^{4} + 46 a^{3} + 67 a^{2} + 39 a + 31\right)\cdot 103^{6} + \left(71 a^{4} + 92 a^{3} + 35 a^{2} + 64 a + 97\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 11 }$ | $=$ | \( 82 a^{4} + 45 a^{3} + 94 a^{2} + 78 a + 48 + \left(18 a^{4} + 43 a^{3} + 42 a + 88\right)\cdot 103 + \left(46 a^{4} + 13 a^{3} + 35 a^{2} + 4 a + 55\right)\cdot 103^{2} + \left(22 a^{4} + 37 a^{3} + 74 a^{2} + 79 a + 36\right)\cdot 103^{3} + \left(26 a^{4} + 95 a^{3} + 64 a^{2} + 31 a + 48\right)\cdot 103^{4} + \left(38 a^{4} + 74 a^{3} + 34 a^{2} + 96 a + 31\right)\cdot 103^{5} + \left(52 a^{4} + 32 a^{3} + 9 a^{2} + 59 a + 31\right)\cdot 103^{6} + \left(41 a^{4} + 11 a^{3} + 100 a^{2} + 59 a + 57\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 12 }$ | $=$ | \( 83 a^{4} + 27 a^{3} + 30 a^{2} + 9 a + 98 + \left(37 a^{4} + 6 a^{3} + 19 a^{2} + 45 a + 90\right)\cdot 103 + \left(13 a^{4} + 94 a^{3} + 16 a^{2} + 99 a + 34\right)\cdot 103^{2} + \left(10 a^{4} + 25 a^{3} + 35 a^{2} + 72\right)\cdot 103^{3} + \left(87 a^{4} + 62 a^{3} + 58 a^{2} + 100 a + 89\right)\cdot 103^{4} + \left(98 a^{4} + 41 a^{3} + 64 a^{2} + 11 a + 90\right)\cdot 103^{5} + \left(5 a^{4} + 15 a^{3} + 37 a^{2} + 47 a + 34\right)\cdot 103^{6} + \left(69 a^{4} + 50 a^{3} + 73 a^{2} + 8 a + 11\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 13 }$ | $=$ | \( 85 a^{4} + 39 a^{3} + 13 a^{2} + 54 a + 61 + \left(32 a^{4} + 56 a^{3} + 73 a^{2} + 7 a + 5\right)\cdot 103 + \left(62 a^{4} + 58 a^{3} + 27 a^{2} + 47 a + 84\right)\cdot 103^{2} + \left(45 a^{4} + 49 a^{3} + 73 a^{2} + 21 a + 94\right)\cdot 103^{3} + \left(63 a^{4} + 22 a^{3} + 85 a^{2} + 63 a + 24\right)\cdot 103^{4} + \left(76 a^{4} + 92 a^{2} + 81 a + 69\right)\cdot 103^{5} + \left(79 a^{4} + 46 a^{3} + 99 a^{2} + 23 a + 12\right)\cdot 103^{6} + \left(44 a^{4} + 93 a^{3} + 60 a^{2} + 3 a + 4\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 14 }$ | $=$ | \( 99 a^{4} + 60 a^{3} + 16 a^{2} + 6 a + 40 + \left(13 a^{4} + 26 a^{3} + 33 a^{2} + 39 a + 4\right)\cdot 103 + \left(74 a^{4} + 32 a^{3} + 45 a^{2} + 57 a + 85\right)\cdot 103^{2} + \left(9 a^{4} + 76 a^{3} + 100 a^{2} + 102 a + 46\right)\cdot 103^{3} + \left(34 a^{4} + 44 a^{3} + 100 a^{2} + 7 a + 34\right)\cdot 103^{4} + \left(69 a^{4} + 31 a^{3} + 81 a^{2} + 51 a + 46\right)\cdot 103^{5} + \left(18 a^{4} + 78 a^{3} + 38 a^{2} + 75 a + 31\right)\cdot 103^{6} + \left(46 a^{4} + 14 a^{3} + 40 a^{2} + 34 a + 78\right)\cdot 103^{7} +O(103^{8})\) |
$r_{ 15 }$ | $=$ | \( 102 a^{4} + 84 a^{3} + a^{2} + 29 a + 55 + \left(63 a^{4} + 85 a^{3} + 56 a^{2} + 22 a + 37\right)\cdot 103 + \left(53 a^{4} + 86 a^{3} + 94 a^{2} + 37 a + 71\right)\cdot 103^{2} + \left(8 a^{4} + 69 a^{3} + 10 a^{2} + 43 a + 68\right)\cdot 103^{3} + \left(87 a^{4} + 41 a^{3} + 86 a^{2} + 20 a + 60\right)\cdot 103^{4} + \left(13 a^{4} + 92 a^{3} + 16 a^{2} + 68 a + 22\right)\cdot 103^{5} + \left(73 a^{3} + 10 a^{2} + 79 a + 69\right)\cdot 103^{6} + \left(84 a^{4} + 38 a^{3} + 12 a^{2} + 47 a + 15\right)\cdot 103^{7} +O(103^{8})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 15 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 15 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$3$ | $2$ | $(1,4)(2,15)(6,13)(8,10)(9,14)$ | $0$ |
$2$ | $3$ | $(1,4,7)(2,5,15)(3,9,14)(6,13,11)(8,10,12)$ | $-1$ |
$1$ | $5$ | $(1,8,9,6,15)(2,4,10,14,13)(3,11,5,7,12)$ | $-2 \zeta_{5}^{3} - 2 \zeta_{5}^{2} - 2 \zeta_{5} - 2$ |
$1$ | $5$ | $(1,9,15,8,6)(2,10,13,4,14)(3,5,12,11,7)$ | $2 \zeta_{5}^{3}$ |
$1$ | $5$ | $(1,6,8,15,9)(2,14,4,13,10)(3,7,11,12,5)$ | $2 \zeta_{5}^{2}$ |
$1$ | $5$ | $(1,15,6,9,8)(2,13,14,10,4)(3,12,7,5,11)$ | $2 \zeta_{5}$ |
$3$ | $10$ | $(1,13,8,2,9,4,6,10,15,14)(3,7,11,12,5)$ | $0$ |
$3$ | $10$ | $(1,2,6,14,8,4,15,13,9,10)(3,12,7,5,11)$ | $0$ |
$3$ | $10$ | $(1,10,9,13,15,4,8,14,6,2)(3,11,5,7,12)$ | $0$ |
$3$ | $10$ | $(1,14,15,10,6,4,9,2,8,13)(3,5,12,11,7)$ | $0$ |
$2$ | $15$ | $(1,11,10,15,3,4,6,12,2,9,7,13,8,5,14)$ | $-\zeta_{5}^{2}$ |
$2$ | $15$ | $(1,10,3,6,2,7,8,14,11,15,4,12,9,13,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + \zeta_{5} + 1$ |
$2$ | $15$ | $(1,3,2,8,11,4,9,5,10,6,7,14,15,12,13)$ | $-\zeta_{5}^{3}$ |
$2$ | $15$ | $(1,2,11,9,10,7,15,13,3,8,4,5,6,14,12)$ | $-\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.