LMFDB - Artin representation 2.579.16t60.a.d

Basic invariants

Dimension:	$2$
Group:	16T60
Conductor:	$579$$\medspace = 3 \cdot 193 $
Artin stem field:	Galois closure of 16.0.12630731694101401542561.1
Galois orbit size:	$4$
Smallest permutation container:	16T60
Parity:	odd
Determinant:	1.579.6t1.a.b
Projective image:	$A_4$
Projective stem field:	Galois closure of 4.0.335241.1

Defining polynomial

$f(x)$

$=$

$ x^{16} - 18x^{12} + 101x^{10} + 99x^{8} - 1098x^{6} + 3043x^{4} - 738x^{2} + 81 $

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^{6} + 92x^{3} + 58x^{2} + 88x + 5 $ x^6 + 92*x^3 + 58*x^2 + 88*x + 5

Roots:

$r_{ 1 }$	$=$	$ 89 a^{5} + 58 a^{4} + 56 a^{3} + 15 a^{2} + 66 a + 61 + \left(68 a^{5} + 21 a^{4} + 49 a^{3} + 87 a^{2} + 20 a + 21\right)\cdot 97 + \left(47 a^{5} + 57 a^{4} + 24 a^{3} + 55 a^{2} + 8 a + 81\right)\cdot 97^{2} + \left(13 a^{5} + 78 a^{4} + 63 a^{3} + 67 a^{2} + 51 a + 50\right)\cdot 97^{3} + \left(25 a^{5} + 28 a^{4} + 13 a^{3} + 67 a^{2} + 49 a + 93\right)\cdot 97^{4} + \left(36 a^{5} + 16 a^{4} + 83 a^{3} + 59 a^{2} + 56 a + 64\right)\cdot 97^{5} + \left(93 a^{5} + 63 a^{4} + 47 a^{3} + 51 a^{2} + 71 a + 77\right)\cdot 97^{6} + \left(27 a^{5} + 76 a^{4} + 59 a^{3} + 91 a^{2} + 79 a + 55\right)\cdot 97^{7} + \left(71 a^{5} + 3 a^{4} + 38 a^{3} + 48 a^{2} + 18 a + 69\right)\cdot 97^{8} + \left(14 a^{5} + 50 a^{4} + 65 a^{3} + 13 a^{2} + 70 a + 9\right)\cdot 97^{9} +O(97^{10})$ 89a^5 + 58a^4 + 56a^3 + 15a^2 + 66a + 61 + (68a^5 + 21a^4 + 49a^3 + 87a^2 + 20a + 21)97 + (47a^5 + 57a^4 + 24a^3 + 55a^2 + 8a + 81)97^2 + (13a^5 + 78a^4 + 63a^3 + 67a^2 + 51a + 50)97^3 + (25a^5 + 28a^4 + 13a^3 + 67a^2 + 49a + 93)97^4 + (36a^5 + 16a^4 + 83a^3 + 59a^2 + 56a + 64)97^5 + (93a^5 + 63a^4 + 47a^3 + 51a^2 + 71a + 77)97^6 + (27a^5 + 76a^4 + 59a^3 + 91a^2 + 79a + 55)97^7 + (71a^5 + 3a^4 + 38a^3 + 48a^2 + 18a + 69)97^8 + (14a^5 + 50a^4 + 65a^3 + 13a^2 + 70a + 9)*97^9+O(97^10)
$r_{ 2 }$	$=$	$ 28 a^{5} + 10 a^{4} + 20 a^{3} + 65 a^{2} + 20 a + 62 + \left(59 a^{5} + 25 a^{4} + 20 a^{3} + a^{2} + 90 a + 7\right)\cdot 97 + \left(19 a^{5} + 43 a^{4} + 10 a^{3} + 53 a^{2} + 13 a + 56\right)\cdot 97^{2} + \left(93 a^{5} + 89 a^{4} + 62 a^{3} + 5 a^{2} + 63 a + 87\right)\cdot 97^{3} + \left(82 a^{5} + 43 a^{4} + 43 a^{3} + 86 a^{2} + 21 a + 24\right)\cdot 97^{4} + \left(41 a^{5} + 7 a^{4} + 80 a^{3} + 93 a^{2} + 74 a + 74\right)\cdot 97^{5} + \left(14 a^{5} + 60 a^{4} + 18 a^{3} + 93 a^{2} + 70 a + 11\right)\cdot 97^{6} + \left(50 a^{5} + 81 a^{4} + 57 a^{3} + 89 a^{2} + 93 a + 23\right)\cdot 97^{7} + \left(40 a^{5} + 6 a^{4} + 32 a^{3} + 56 a^{2} + 49 a + 14\right)\cdot 97^{8} + \left(83 a^{5} + 8 a^{4} + 95 a^{3} + 64 a^{2} + 68 a + 80\right)\cdot 97^{9} +O(97^{10})$ 28a^5 + 10a^4 + 20a^3 + 65a^2 + 20a + 62 + (59a^5 + 25a^4 + 20a^3 + a^2 + 90a + 7)97 + (19a^5 + 43a^4 + 10a^3 + 53a^2 + 13a + 56)97^2 + (93a^5 + 89a^4 + 62a^3 + 5a^2 + 63a + 87)97^3 + (82a^5 + 43a^4 + 43a^3 + 86a^2 + 21a + 24)97^4 + (41a^5 + 7a^4 + 80a^3 + 93a^2 + 74a + 74)97^5 + (14a^5 + 60a^4 + 18a^3 + 93a^2 + 70a + 11)97^6 + (50a^5 + 81a^4 + 57a^3 + 89a^2 + 93a + 23)97^7 + (40a^5 + 6a^4 + 32a^3 + 56a^2 + 49a + 14)97^8 + (83a^5 + 8a^4 + 95a^3 + 64a^2 + 68a + 80)97^9+O(97^10)
$r_{ 3 }$	$=$	$ 20 a^{5} + 31 a^{4} + 84 a^{3} + 3 a^{2} + 39 a + 95 + \left(10 a^{5} + 93 a^{4} + 14 a^{3} + 67 a^{2} + 9 a + 28\right)\cdot 97 + \left(22 a^{5} + 71 a^{4} + 10 a^{3} + 63 a^{2} + 84 a + 85\right)\cdot 97^{2} + \left(90 a^{5} + 5 a^{4} + 79 a^{3} + 47 a^{2} + 24 a + 33\right)\cdot 97^{3} + \left(92 a^{5} + 38 a^{4} + 79 a^{3} + 12 a^{2} + 34 a + 59\right)\cdot 97^{4} + \left(93 a^{5} + 44 a^{4} + 35 a^{3} + 95 a^{2} + 89 a + 69\right)\cdot 97^{5} + \left(55 a^{5} + 51 a^{4} + 27 a^{3} + 90 a^{2} + 5 a + 45\right)\cdot 97^{6} + \left(71 a^{5} + 18 a^{4} + 82 a^{3} + 65 a^{2} + 12 a + 50\right)\cdot 97^{7} + \left(72 a^{5} + 17 a^{4} + 84 a^{3} + 10 a^{2} + 16 a + 72\right)\cdot 97^{8} + \left(60 a^{5} + 96 a^{4} + 84 a^{3} + 7 a^{2} + 19 a + 17\right)\cdot 97^{9} +O(97^{10})$ 20a^5 + 31a^4 + 84a^3 + 3a^2 + 39a + 95 + (10a^5 + 93a^4 + 14a^3 + 67a^2 + 9a + 28)97 + (22a^5 + 71a^4 + 10a^3 + 63a^2 + 84a + 85)97^2 + (90a^5 + 5a^4 + 79a^3 + 47a^2 + 24a + 33)97^3 + (92a^5 + 38a^4 + 79a^3 + 12a^2 + 34a + 59)97^4 + (93a^5 + 44a^4 + 35a^3 + 95a^2 + 89a + 69)97^5 + (55a^5 + 51a^4 + 27a^3 + 90a^2 + 5a + 45)97^6 + (71a^5 + 18a^4 + 82a^3 + 65a^2 + 12a + 50)97^7 + (72a^5 + 17a^4 + 84a^3 + 10a^2 + 16a + 72)97^8 + (60a^5 + 96a^4 + 84a^3 + 7a^2 + 19a + 17)*97^9+O(97^10)
$r_{ 4 }$	$=$	$ 38 a^{5} + 10 a^{4} + 26 a^{3} + 77 a^{2} + 49 a + 69 + \left(6 a^{5} + 29 a^{4} + 58 a^{3} + 81 a^{2} + 31 a + 38\right)\cdot 97 + \left(10 a^{5} + 11 a^{4} + 32 a^{3} + 52 a^{2} + 41 a + 23\right)\cdot 97^{2} + \left(53 a^{5} + 35 a^{4} + 11 a^{3} + 38 a^{2} + 44 a + 18\right)\cdot 97^{3} + \left(56 a^{5} + 26 a^{4} + 18 a^{3} + 64 a^{2} + 4 a + 50\right)\cdot 97^{4} + \left(64 a^{5} + 60 a^{4} + 53 a^{3} + 94 a^{2} + 94 a + 39\right)\cdot 97^{5} + \left(53 a^{5} + 10 a^{4} + 31 a^{3} + 33 a^{2} + 96 a + 74\right)\cdot 97^{6} + \left(74 a^{5} + 28 a^{4} + 63 a^{3} + 63 a^{2} + 9 a + 25\right)\cdot 97^{7} + \left(80 a^{5} + 39 a^{4} + 8 a^{3} + 79 a^{2} + 62 a + 80\right)\cdot 97^{8} + \left(21 a^{5} + 44 a^{4} + 15 a^{3} + 78 a^{2} + 4 a + 2\right)\cdot 97^{9} +O(97^{10})$ 38a^5 + 10a^4 + 26a^3 + 77a^2 + 49a + 69 + (6a^5 + 29a^4 + 58a^3 + 81a^2 + 31a + 38)97 + (10a^5 + 11a^4 + 32a^3 + 52a^2 + 41a + 23)97^2 + (53a^5 + 35a^4 + 11a^3 + 38a^2 + 44a + 18)97^3 + (56a^5 + 26a^4 + 18a^3 + 64a^2 + 4a + 50)97^4 + (64a^5 + 60a^4 + 53a^3 + 94a^2 + 94a + 39)97^5 + (53a^5 + 10a^4 + 31a^3 + 33a^2 + 96a + 74)97^6 + (74a^5 + 28a^4 + 63a^3 + 63a^2 + 9a + 25)97^7 + (80a^5 + 39a^4 + 8a^3 + 79a^2 + 62a + 80)97^8 + (21a^5 + 44a^4 + 15a^3 + 78a^2 + 4a + 2)*97^9+O(97^10)
$r_{ 5 }$	$=$	$ 22 a^{5} + 16 a^{4} + 61 a^{3} + 65 a^{2} + 59 a + 91 + \left(78 a^{5} + 50 a^{4} + 36 a^{3} + 26 a^{2} + 17 a + 16\right)\cdot 97 + \left(32 a^{5} + 11 a^{4} + 82 a^{3} + 12 a^{2} + 45 a + 12\right)\cdot 97^{2} + \left(11 a^{5} + 59 a^{4} + 70 a^{3} + 15 a^{2} + 21 a + 39\right)\cdot 97^{3} + \left(73 a^{5} + 92 a^{4} + 53 a^{3} + 61 a^{2} + 41 a + 65\right)\cdot 97^{4} + \left(39 a^{5} + 91 a^{4} + 82 a^{3} + 61 a^{2} + 5 a + 18\right)\cdot 97^{5} + \left(15 a^{5} + 84 a^{4} + 70 a^{3} + 3 a^{2} + 49 a + 43\right)\cdot 97^{6} + \left(87 a^{5} + 72 a^{4} + 5 a^{3} + 26 a^{2} + 37 a + 96\right)\cdot 97^{7} + \left(58 a^{5} + 96 a^{4} + 55 a^{3} + 76 a^{2} + 35 a + 19\right)\cdot 97^{8} + \left(65 a^{5} + 6 a^{4} + 24 a^{3} + 6 a^{2} + 68 a + 36\right)\cdot 97^{9} +O(97^{10})$ 22a^5 + 16a^4 + 61a^3 + 65a^2 + 59a + 91 + (78a^5 + 50a^4 + 36a^3 + 26a^2 + 17a + 16)97 + (32a^5 + 11a^4 + 82a^3 + 12a^2 + 45a + 12)97^2 + (11a^5 + 59a^4 + 70a^3 + 15a^2 + 21a + 39)97^3 + (73a^5 + 92a^4 + 53a^3 + 61a^2 + 41a + 65)97^4 + (39a^5 + 91a^4 + 82a^3 + 61a^2 + 5a + 18)97^5 + (15a^5 + 84a^4 + 70a^3 + 3a^2 + 49a + 43)97^6 + (87a^5 + 72a^4 + 5a^3 + 26a^2 + 37a + 96)97^7 + (58a^5 + 96a^4 + 55a^3 + 76a^2 + 35a + 19)97^8 + (65a^5 + 6a^4 + 24a^3 + 6a^2 + 68a + 36)*97^9+O(97^10)
$r_{ 6 }$	$=$	$ 9 a^{5} + 73 a^{4} + 75 a^{3} + 11 a^{2} + 20 a + 78 + \left(19 a^{5} + 12 a^{4} + 63 a^{3} + 45 a^{2} + 83 a + 73\right)\cdot 97 + \left(32 a^{5} + 60 a^{4} + 26 a^{3} + 46 a^{2} + 76 a + 27\right)\cdot 97^{2} + \left(93 a^{5} + 19 a^{4} + 21 a^{3} + 3 a^{2} + 82 a + 13\right)\cdot 97^{3} + \left(64 a^{5} + 78 a^{4} + 93 a^{3} + 95 a^{2} + 89 a + 80\right)\cdot 97^{4} + \left(26 a^{5} + 18 a^{4} + 26 a^{3} + 77 a^{2} + 36 a + 10\right)\cdot 97^{5} + \left(2 a^{5} + 47 a^{4} + 92 a^{3} + 77 a^{2} + 35 a + 93\right)\cdot 97^{6} + \left(24 a^{5} + 60 a^{4} + 17 a^{3} + 61 a^{2} + 72 a + 43\right)\cdot 97^{7} + \left(55 a^{5} + 23 a^{4} + 26 a^{3} + 27 a^{2} + 58\right)\cdot 97^{8} + \left(24 a^{5} + 4 a^{4} + 29 a^{3} + a^{2} + 95 a + 46\right)\cdot 97^{9} +O(97^{10})$ 9a^5 + 73a^4 + 75a^3 + 11a^2 + 20a + 78 + (19a^5 + 12a^4 + 63a^3 + 45a^2 + 83a + 73)97 + (32a^5 + 60a^4 + 26a^3 + 46a^2 + 76a + 27)97^2 + (93a^5 + 19a^4 + 21a^3 + 3a^2 + 82a + 13)97^3 + (64a^5 + 78a^4 + 93a^3 + 95a^2 + 89a + 80)97^4 + (26a^5 + 18a^4 + 26a^3 + 77a^2 + 36a + 10)97^5 + (2a^5 + 47a^4 + 92a^3 + 77a^2 + 35a + 93)97^6 + (24a^5 + 60a^4 + 17a^3 + 61a^2 + 72a + 43)97^7 + (55a^5 + 23a^4 + 26a^3 + 27a^2 + 58)97^8 + (24a^5 + 4a^4 + 29a^3 + a^2 + 95a + 46)*97^9+O(97^10)
$r_{ 7 }$	$=$	$ 31 a^{5} + 87 a^{4} + 93 a^{3} + 64 a^{2} + 25 a + 86 + \left(34 a^{5} + 17 a^{4} + 53 a^{3} + 92 a^{2} + 52 a + 64\right)\cdot 97 + \left(88 a^{5} + 4 a^{4} + 74 a^{3} + 96 a^{2} + 96 a + 79\right)\cdot 97^{2} + \left(59 a^{5} + 4 a^{4} + 61 a^{3} + 3 a^{2} + 30 a + 83\right)\cdot 97^{3} + \left(50 a^{5} + 9 a^{4} + 21 a^{3} + 28 a^{2} + 73 a + 12\right)\cdot 97^{4} + \left(57 a^{5} + 16 a^{4} + 56 a^{3} + 49 a^{2} + 12 a + 87\right)\cdot 97^{5} + \left(28 a^{5} + 80 a^{4} + 9 a^{3} + 19 a^{2} + 85 a + 75\right)\cdot 97^{6} + \left(72 a^{5} + 87 a^{4} + 19 a^{3} + 38 a^{2} + 55 a + 37\right)\cdot 97^{7} + \left(71 a^{5} + 57 a^{4} + 43 a^{3} + 37 a^{2} + 85 a + 78\right)\cdot 97^{8} + \left(82 a^{5} + 4 a^{4} + 13 a^{3} + 57 a^{2} + 32 a + 73\right)\cdot 97^{9} +O(97^{10})$ 31a^5 + 87a^4 + 93a^3 + 64a^2 + 25a + 86 + (34a^5 + 17a^4 + 53a^3 + 92a^2 + 52a + 64)97 + (88a^5 + 4a^4 + 74a^3 + 96a^2 + 96a + 79)97^2 + (59a^5 + 4a^4 + 61a^3 + 3a^2 + 30a + 83)97^3 + (50a^5 + 9a^4 + 21a^3 + 28a^2 + 73a + 12)97^4 + (57a^5 + 16a^4 + 56a^3 + 49a^2 + 12a + 87)97^5 + (28a^5 + 80a^4 + 9a^3 + 19a^2 + 85a + 75)97^6 + (72a^5 + 87a^4 + 19a^3 + 38a^2 + 55a + 37)97^7 + (71a^5 + 57a^4 + 43a^3 + 37a^2 + 85a + 78)97^8 + (82a^5 + 4a^4 + 13a^3 + 57a^2 + 32a + 73)*97^9+O(97^10)
$r_{ 8 }$	$=$	$ 27 a^{5} + 84 a^{4} + 82 a^{3} + 52 a^{2} + 69 a + 76 + \left(94 a^{5} + 76 a^{4} + 56 a^{3} + 73 a^{2} + 13 a + 84\right)\cdot 97 + \left(68 a^{5} + 80 a^{4} + 50 a^{3} + 43 a^{2} + 5 a + 32\right)\cdot 97^{2} + \left(19 a^{5} + 21 a^{4} + 96 a^{3} + 61 a^{2} + 73 a + 4\right)\cdot 97^{3} + \left(72 a^{5} + 26 a^{4} + 80 a^{3} + 4 a^{2} + 24 a + 76\right)\cdot 97^{4} + \left(36 a^{5} + 6 a^{4} + 52 a^{3} + 34 a^{2} + 21 a + 64\right)\cdot 97^{5} + \left(27 a^{5} + 29 a^{4} + 69 a^{3} + 17 a^{2} + 6 a + 25\right)\cdot 97^{6} + \left(4 a^{5} + 22 a^{4} + 31 a^{3} + 89 a^{2} + 5 a + 16\right)\cdot 97^{7} + \left(22 a^{5} + 77 a^{4} + 26 a^{3} + 70 a^{2} + 91 a + 26\right)\cdot 97^{8} + \left(93 a^{5} + 34 a^{4} + a^{3} + 68 a^{2} + 35 a + 60\right)\cdot 97^{9} +O(97^{10})$ 27a^5 + 84a^4 + 82a^3 + 52a^2 + 69a + 76 + (94a^5 + 76a^4 + 56a^3 + 73a^2 + 13a + 84)97 + (68a^5 + 80a^4 + 50a^3 + 43a^2 + 5a + 32)97^2 + (19a^5 + 21a^4 + 96a^3 + 61a^2 + 73a + 4)97^3 + (72a^5 + 26a^4 + 80a^3 + 4a^2 + 24a + 76)97^4 + (36a^5 + 6a^4 + 52a^3 + 34a^2 + 21a + 64)97^5 + (27a^5 + 29a^4 + 69a^3 + 17a^2 + 6a + 25)97^6 + (4a^5 + 22a^4 + 31a^3 + 89a^2 + 5a + 16)97^7 + (22a^5 + 77a^4 + 26a^3 + 70a^2 + 91a + 26)97^8 + (93a^5 + 34a^4 + a^3 + 68a^2 + 35a + 60)97^9+O(97^10)
$r_{ 9 }$	$=$	$ 8 a^{5} + 39 a^{4} + 41 a^{3} + 82 a^{2} + 31 a + 36 + \left(28 a^{5} + 75 a^{4} + 47 a^{3} + 9 a^{2} + 76 a + 75\right)\cdot 97 + \left(49 a^{5} + 39 a^{4} + 72 a^{3} + 41 a^{2} + 88 a + 15\right)\cdot 97^{2} + \left(83 a^{5} + 18 a^{4} + 33 a^{3} + 29 a^{2} + 45 a + 46\right)\cdot 97^{3} + \left(71 a^{5} + 68 a^{4} + 83 a^{3} + 29 a^{2} + 47 a + 3\right)\cdot 97^{4} + \left(60 a^{5} + 80 a^{4} + 13 a^{3} + 37 a^{2} + 40 a + 32\right)\cdot 97^{5} + \left(3 a^{5} + 33 a^{4} + 49 a^{3} + 45 a^{2} + 25 a + 19\right)\cdot 97^{6} + \left(69 a^{5} + 20 a^{4} + 37 a^{3} + 5 a^{2} + 17 a + 41\right)\cdot 97^{7} + \left(25 a^{5} + 93 a^{4} + 58 a^{3} + 48 a^{2} + 78 a + 27\right)\cdot 97^{8} + \left(82 a^{5} + 46 a^{4} + 31 a^{3} + 83 a^{2} + 26 a + 87\right)\cdot 97^{9} +O(97^{10})$ 8a^5 + 39a^4 + 41a^3 + 82a^2 + 31a + 36 + (28a^5 + 75a^4 + 47a^3 + 9a^2 + 76a + 75)97 + (49a^5 + 39a^4 + 72a^3 + 41a^2 + 88a + 15)97^2 + (83a^5 + 18a^4 + 33a^3 + 29a^2 + 45a + 46)97^3 + (71a^5 + 68a^4 + 83a^3 + 29a^2 + 47a + 3)97^4 + (60a^5 + 80a^4 + 13a^3 + 37a^2 + 40a + 32)97^5 + (3a^5 + 33a^4 + 49a^3 + 45a^2 + 25a + 19)97^6 + (69a^5 + 20a^4 + 37a^3 + 5a^2 + 17a + 41)97^7 + (25a^5 + 93a^4 + 58a^3 + 48a^2 + 78a + 27)97^8 + (82a^5 + 46a^4 + 31a^3 + 83a^2 + 26a + 87)*97^9+O(97^10)
$r_{ 10 }$	$=$	$ 69 a^{5} + 87 a^{4} + 77 a^{3} + 32 a^{2} + 77 a + 35 + \left(37 a^{5} + 71 a^{4} + 76 a^{3} + 95 a^{2} + 6 a + 89\right)\cdot 97 + \left(77 a^{5} + 53 a^{4} + 86 a^{3} + 43 a^{2} + 83 a + 40\right)\cdot 97^{2} + \left(3 a^{5} + 7 a^{4} + 34 a^{3} + 91 a^{2} + 33 a + 9\right)\cdot 97^{3} + \left(14 a^{5} + 53 a^{4} + 53 a^{3} + 10 a^{2} + 75 a + 72\right)\cdot 97^{4} + \left(55 a^{5} + 89 a^{4} + 16 a^{3} + 3 a^{2} + 22 a + 22\right)\cdot 97^{5} + \left(82 a^{5} + 36 a^{4} + 78 a^{3} + 3 a^{2} + 26 a + 85\right)\cdot 97^{6} + \left(46 a^{5} + 15 a^{4} + 39 a^{3} + 7 a^{2} + 3 a + 73\right)\cdot 97^{7} + \left(56 a^{5} + 90 a^{4} + 64 a^{3} + 40 a^{2} + 47 a + 82\right)\cdot 97^{8} + \left(13 a^{5} + 88 a^{4} + a^{3} + 32 a^{2} + 28 a + 16\right)\cdot 97^{9} +O(97^{10})$ 69a^5 + 87a^4 + 77a^3 + 32a^2 + 77a + 35 + (37a^5 + 71a^4 + 76a^3 + 95a^2 + 6a + 89)97 + (77a^5 + 53a^4 + 86a^3 + 43a^2 + 83a + 40)97^2 + (3a^5 + 7a^4 + 34a^3 + 91a^2 + 33a + 9)97^3 + (14a^5 + 53a^4 + 53a^3 + 10a^2 + 75a + 72)97^4 + (55a^5 + 89a^4 + 16a^3 + 3a^2 + 22a + 22)97^5 + (82a^5 + 36a^4 + 78a^3 + 3a^2 + 26a + 85)97^6 + (46a^5 + 15a^4 + 39a^3 + 7a^2 + 3a + 73)97^7 + (56a^5 + 90a^4 + 64a^3 + 40a^2 + 47a + 82)97^8 + (13a^5 + 88a^4 + a^3 + 32a^2 + 28a + 16)97^9+O(97^10)
$r_{ 11 }$	$=$	$ 77 a^{5} + 66 a^{4} + 13 a^{3} + 94 a^{2} + 58 a + 2 + \left(86 a^{5} + 3 a^{4} + 82 a^{3} + 29 a^{2} + 87 a + 68\right)\cdot 97 + \left(74 a^{5} + 25 a^{4} + 86 a^{3} + 33 a^{2} + 12 a + 11\right)\cdot 97^{2} + \left(6 a^{5} + 91 a^{4} + 17 a^{3} + 49 a^{2} + 72 a + 63\right)\cdot 97^{3} + \left(4 a^{5} + 58 a^{4} + 17 a^{3} + 84 a^{2} + 62 a + 37\right)\cdot 97^{4} + \left(3 a^{5} + 52 a^{4} + 61 a^{3} + a^{2} + 7 a + 27\right)\cdot 97^{5} + \left(41 a^{5} + 45 a^{4} + 69 a^{3} + 6 a^{2} + 91 a + 51\right)\cdot 97^{6} + \left(25 a^{5} + 78 a^{4} + 14 a^{3} + 31 a^{2} + 84 a + 46\right)\cdot 97^{7} + \left(24 a^{5} + 79 a^{4} + 12 a^{3} + 86 a^{2} + 80 a + 24\right)\cdot 97^{8} + \left(36 a^{5} + 12 a^{3} + 89 a^{2} + 77 a + 79\right)\cdot 97^{9} +O(97^{10})$ 77a^5 + 66a^4 + 13a^3 + 94a^2 + 58a + 2 + (86a^5 + 3a^4 + 82a^3 + 29a^2 + 87a + 68)97 + (74a^5 + 25a^4 + 86a^3 + 33a^2 + 12a + 11)97^2 + (6a^5 + 91a^4 + 17a^3 + 49a^2 + 72a + 63)97^3 + (4a^5 + 58a^4 + 17a^3 + 84a^2 + 62a + 37)97^4 + (3a^5 + 52a^4 + 61a^3 + a^2 + 7a + 27)97^5 + (41a^5 + 45a^4 + 69a^3 + 6a^2 + 91a + 51)97^6 + (25a^5 + 78a^4 + 14a^3 + 31a^2 + 84a + 46)97^7 + (24a^5 + 79a^4 + 12a^3 + 86a^2 + 80a + 24)97^8 + (36a^5 + 12a^3 + 89a^2 + 77a + 79)*97^9+O(97^10)
$r_{ 12 }$	$=$	$ 59 a^{5} + 87 a^{4} + 71 a^{3} + 20 a^{2} + 48 a + 28 + \left(90 a^{5} + 67 a^{4} + 38 a^{3} + 15 a^{2} + 65 a + 58\right)\cdot 97 + \left(86 a^{5} + 85 a^{4} + 64 a^{3} + 44 a^{2} + 55 a + 73\right)\cdot 97^{2} + \left(43 a^{5} + 61 a^{4} + 85 a^{3} + 58 a^{2} + 52 a + 78\right)\cdot 97^{3} + \left(40 a^{5} + 70 a^{4} + 78 a^{3} + 32 a^{2} + 92 a + 46\right)\cdot 97^{4} + \left(32 a^{5} + 36 a^{4} + 43 a^{3} + 2 a^{2} + 2 a + 57\right)\cdot 97^{5} + \left(43 a^{5} + 86 a^{4} + 65 a^{3} + 63 a^{2} + 22\right)\cdot 97^{6} + \left(22 a^{5} + 68 a^{4} + 33 a^{3} + 33 a^{2} + 87 a + 71\right)\cdot 97^{7} + \left(16 a^{5} + 57 a^{4} + 88 a^{3} + 17 a^{2} + 34 a + 16\right)\cdot 97^{8} + \left(75 a^{5} + 52 a^{4} + 81 a^{3} + 18 a^{2} + 92 a + 94\right)\cdot 97^{9} +O(97^{10})$ 59a^5 + 87a^4 + 71a^3 + 20a^2 + 48a + 28 + (90a^5 + 67a^4 + 38a^3 + 15a^2 + 65a + 58)97 + (86a^5 + 85a^4 + 64a^3 + 44a^2 + 55a + 73)97^2 + (43a^5 + 61a^4 + 85a^3 + 58a^2 + 52a + 78)97^3 + (40a^5 + 70a^4 + 78a^3 + 32a^2 + 92a + 46)97^4 + (32a^5 + 36a^4 + 43a^3 + 2a^2 + 2a + 57)97^5 + (43a^5 + 86a^4 + 65a^3 + 63a^2 + 22)97^6 + (22a^5 + 68a^4 + 33a^3 + 33a^2 + 87a + 71)97^7 + (16a^5 + 57a^4 + 88a^3 + 17a^2 + 34a + 16)97^8 + (75a^5 + 52a^4 + 81a^3 + 18a^2 + 92a + 94)97^9+O(97^10)
$r_{ 13 }$	$=$	$ 75 a^{5} + 81 a^{4} + 36 a^{3} + 32 a^{2} + 38 a + 6 + \left(18 a^{5} + 46 a^{4} + 60 a^{3} + 70 a^{2} + 79 a + 80\right)\cdot 97 + \left(64 a^{5} + 85 a^{4} + 14 a^{3} + 84 a^{2} + 51 a + 84\right)\cdot 97^{2} + \left(85 a^{5} + 37 a^{4} + 26 a^{3} + 81 a^{2} + 75 a + 57\right)\cdot 97^{3} + \left(23 a^{5} + 4 a^{4} + 43 a^{3} + 35 a^{2} + 55 a + 31\right)\cdot 97^{4} + \left(57 a^{5} + 5 a^{4} + 14 a^{3} + 35 a^{2} + 91 a + 78\right)\cdot 97^{5} + \left(81 a^{5} + 12 a^{4} + 26 a^{3} + 93 a^{2} + 47 a + 53\right)\cdot 97^{6} + \left(9 a^{5} + 24 a^{4} + 91 a^{3} + 70 a^{2} + 59 a\right)\cdot 97^{7} + \left(38 a^{5} + 41 a^{3} + 20 a^{2} + 61 a + 77\right)\cdot 97^{8} + \left(31 a^{5} + 90 a^{4} + 72 a^{3} + 90 a^{2} + 28 a + 60\right)\cdot 97^{9} +O(97^{10})$ 75a^5 + 81a^4 + 36a^3 + 32a^2 + 38a + 6 + (18a^5 + 46a^4 + 60a^3 + 70a^2 + 79a + 80)97 + (64a^5 + 85a^4 + 14a^3 + 84a^2 + 51a + 84)97^2 + (85a^5 + 37a^4 + 26a^3 + 81a^2 + 75a + 57)97^3 + (23a^5 + 4a^4 + 43a^3 + 35a^2 + 55a + 31)97^4 + (57a^5 + 5a^4 + 14a^3 + 35a^2 + 91a + 78)97^5 + (81a^5 + 12a^4 + 26a^3 + 93a^2 + 47a + 53)97^6 + (9a^5 + 24a^4 + 91a^3 + 70a^2 + 59a)97^7 + (38a^5 + 41a^3 + 20a^2 + 61a + 77)97^8 + (31a^5 + 90a^4 + 72a^3 + 90a^2 + 28a + 60)97^9+O(97^10)
$r_{ 14 }$	$=$	$ 88 a^{5} + 24 a^{4} + 22 a^{3} + 86 a^{2} + 77 a + 19 + \left(77 a^{5} + 84 a^{4} + 33 a^{3} + 51 a^{2} + 13 a + 23\right)\cdot 97 + \left(64 a^{5} + 36 a^{4} + 70 a^{3} + 50 a^{2} + 20 a + 69\right)\cdot 97^{2} + \left(3 a^{5} + 77 a^{4} + 75 a^{3} + 93 a^{2} + 14 a + 83\right)\cdot 97^{3} + \left(32 a^{5} + 18 a^{4} + 3 a^{3} + a^{2} + 7 a + 16\right)\cdot 97^{4} + \left(70 a^{5} + 78 a^{4} + 70 a^{3} + 19 a^{2} + 60 a + 86\right)\cdot 97^{5} + \left(94 a^{5} + 49 a^{4} + 4 a^{3} + 19 a^{2} + 61 a + 3\right)\cdot 97^{6} + \left(72 a^{5} + 36 a^{4} + 79 a^{3} + 35 a^{2} + 24 a + 53\right)\cdot 97^{7} + \left(41 a^{5} + 73 a^{4} + 70 a^{3} + 69 a^{2} + 96 a + 38\right)\cdot 97^{8} + \left(72 a^{5} + 92 a^{4} + 67 a^{3} + 95 a^{2} + a + 50\right)\cdot 97^{9} +O(97^{10})$ 88a^5 + 24a^4 + 22a^3 + 86a^2 + 77a + 19 + (77a^5 + 84a^4 + 33a^3 + 51a^2 + 13a + 23)97 + (64a^5 + 36a^4 + 70a^3 + 50a^2 + 20a + 69)97^2 + (3a^5 + 77a^4 + 75a^3 + 93a^2 + 14a + 83)97^3 + (32a^5 + 18a^4 + 3a^3 + a^2 + 7a + 16)97^4 + (70a^5 + 78a^4 + 70a^3 + 19a^2 + 60a + 86)97^5 + (94a^5 + 49a^4 + 4a^3 + 19a^2 + 61a + 3)97^6 + (72a^5 + 36a^4 + 79a^3 + 35a^2 + 24a + 53)97^7 + (41a^5 + 73a^4 + 70a^3 + 69a^2 + 96a + 38)97^8 + (72a^5 + 92a^4 + 67a^3 + 95a^2 + a + 50)*97^9+O(97^10)
$r_{ 15 }$	$=$	$ 66 a^{5} + 10 a^{4} + 4 a^{3} + 33 a^{2} + 72 a + 11 + \left(62 a^{5} + 79 a^{4} + 43 a^{3} + 4 a^{2} + 44 a + 32\right)\cdot 97 + \left(8 a^{5} + 92 a^{4} + 22 a^{3} + 17\right)\cdot 97^{2} + \left(37 a^{5} + 92 a^{4} + 35 a^{3} + 93 a^{2} + 66 a + 13\right)\cdot 97^{3} + \left(46 a^{5} + 87 a^{4} + 75 a^{3} + 68 a^{2} + 23 a + 84\right)\cdot 97^{4} + \left(39 a^{5} + 80 a^{4} + 40 a^{3} + 47 a^{2} + 84 a + 9\right)\cdot 97^{5} + \left(68 a^{5} + 16 a^{4} + 87 a^{3} + 77 a^{2} + 11 a + 21\right)\cdot 97^{6} + \left(24 a^{5} + 9 a^{4} + 77 a^{3} + 58 a^{2} + 41 a + 59\right)\cdot 97^{7} + \left(25 a^{5} + 39 a^{4} + 53 a^{3} + 59 a^{2} + 11 a + 18\right)\cdot 97^{8} + \left(14 a^{5} + 92 a^{4} + 83 a^{3} + 39 a^{2} + 64 a + 23\right)\cdot 97^{9} +O(97^{10})$ 66a^5 + 10a^4 + 4a^3 + 33a^2 + 72a + 11 + (62a^5 + 79a^4 + 43a^3 + 4a^2 + 44a + 32)97 + (8a^5 + 92a^4 + 22a^3 + 17)97^2 + (37a^5 + 92a^4 + 35a^3 + 93a^2 + 66a + 13)97^3 + (46a^5 + 87a^4 + 75a^3 + 68a^2 + 23a + 84)97^4 + (39a^5 + 80a^4 + 40a^3 + 47a^2 + 84a + 9)97^5 + (68a^5 + 16a^4 + 87a^3 + 77a^2 + 11a + 21)97^6 + (24a^5 + 9a^4 + 77a^3 + 58a^2 + 41a + 59)97^7 + (25a^5 + 39a^4 + 53a^3 + 59a^2 + 11a + 18)97^8 + (14a^5 + 92a^4 + 83a^3 + 39a^2 + 64a + 23)*97^9+O(97^10)
$r_{ 16 }$	$=$	$ 70 a^{5} + 13 a^{4} + 15 a^{3} + 45 a^{2} + 28 a + 21 + \left(2 a^{5} + 20 a^{4} + 40 a^{3} + 23 a^{2} + 83 a + 12\right)\cdot 97 + \left(28 a^{5} + 16 a^{4} + 46 a^{3} + 53 a^{2} + 91 a + 64\right)\cdot 97^{2} + \left(77 a^{5} + 75 a^{4} + 35 a^{2} + 23 a + 92\right)\cdot 97^{3} + \left(24 a^{5} + 70 a^{4} + 16 a^{3} + 92 a^{2} + 72 a + 20\right)\cdot 97^{4} + \left(60 a^{5} + 90 a^{4} + 44 a^{3} + 62 a^{2} + 75 a + 32\right)\cdot 97^{5} + \left(69 a^{5} + 67 a^{4} + 27 a^{3} + 79 a^{2} + 90 a + 71\right)\cdot 97^{6} + \left(92 a^{5} + 74 a^{4} + 65 a^{3} + 7 a^{2} + 91 a + 80\right)\cdot 97^{7} + \left(74 a^{5} + 19 a^{4} + 70 a^{3} + 26 a^{2} + 5 a + 70\right)\cdot 97^{8} + \left(3 a^{5} + 62 a^{4} + 95 a^{3} + 28 a^{2} + 61 a + 36\right)\cdot 97^{9} +O(97^{10})$ 70a^5 + 13a^4 + 15a^3 + 45a^2 + 28a + 21 + (2a^5 + 20a^4 + 40a^3 + 23a^2 + 83a + 12)97 + (28a^5 + 16a^4 + 46a^3 + 53a^2 + 91a + 64)97^2 + (77a^5 + 75a^4 + 35a^2 + 23a + 92)97^3 + (24a^5 + 70a^4 + 16a^3 + 92a^2 + 72a + 20)97^4 + (60a^5 + 90a^4 + 44a^3 + 62a^2 + 75a + 32)97^5 + (69a^5 + 67a^4 + 27a^3 + 79a^2 + 90a + 71)97^6 + (92a^5 + 74a^4 + 65a^3 + 7a^2 + 91a + 80)97^7 + (74a^5 + 19a^4 + 70a^3 + 26a^2 + 5a + 70)97^8 + (3a^5 + 62a^4 + 95a^3 + 28a^2 + 61a + 36)97^9+O(97^10)

Generators of the action on the roots $r_1, \ldots, r_{ 16 }$

Cycle notation
$(1,14,9,6)(2,15,10,7)(3,4,11,12)(5,8,13,16)$
$(1,5,9,13)(2,11,10,3)(4,7,12,15)(6,8,14,16)$
$(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$
$(1,7,9,15)(2,6,10,14)(3,16,11,8)(4,5,12,13)$
$(1,8,14)(2,7,3)(6,9,16)(10,15,11)$

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 16 }$	Character value
$1$	$1$	$()$	$2$
$1$	$2$	$(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$	$-2$
$6$	$2$	$(1,2)(3,5)(4,8)(6,7)(9,10)(11,13)(12,16)(14,15)$	$0$
$4$	$3$	$(2,11,12)(3,4,10)(5,14,16)(6,8,13)$	$\zeta_{12}^{2}$
$4$	$3$	$(2,12,11)(3,10,4)(5,16,14)(6,13,8)$	$-\zeta_{12}^{2} + 1$
$1$	$4$	$(1,7,9,15)(2,6,10,14)(3,16,11,8)(4,5,12,13)$	$-2 \zeta_{12}^{3}$
$1$	$4$	$(1,15,9,7)(2,14,10,6)(3,8,11,16)(4,13,12,5)$	$2 \zeta_{12}^{3}$
$6$	$4$	$(1,14,9,6)(2,15,10,7)(3,4,11,12)(5,8,13,16)$	$0$
$4$	$6$	$(1,9)(2,4,11,10,12,3)(5,8,14,13,16,6)(7,15)$	$\zeta_{12}^{2} - 1$
$4$	$6$	$(1,9)(2,3,12,10,11,4)(5,6,16,13,14,8)(7,15)$	$-\zeta_{12}^{2}$
$4$	$12$	$(1,7,9,15)(2,8,4,14,11,13,10,16,12,6,3,5)$	$-\zeta_{12}^{3} + \zeta_{12}$
$4$	$12$	$(1,7,9,15)(2,13,3,14,12,8,10,5,11,6,4,16)$	$-\zeta_{12}$
$4$	$12$	$(1,15,9,7)(2,16,4,6,11,5,10,8,12,14,3,13)$	$\zeta_{12}^{3} - \zeta_{12}$
$4$	$12$	$(1,15,9,7)(2,5,3,6,12,16,10,13,11,14,4,8)$	$\zeta_{12}$

The blue line marks the conjugacy class containing complex conjugation.

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Generators of the action on the roots $r_1, \ldots, r_{ 16 }$

Character values on conjugacy classes

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	2.579.16t60.a.d
Dimension	$2$
Group	$\SL(2,3):C_2$
Conductor	$579$
Root number	not computed
Indicator	$0$