LMFDB - Artin representation 2.2033.16t60.a.a

Basic invariants

Dimension:	$2$
Group:	16T60
Conductor:	$2033$$\medspace = 19 \cdot 107 $
Artin stem field:	Galois closure of 16.0.291809233013503778024250241.1
Galois orbit size:	$4$
Smallest permutation container:	16T60
Parity:	odd
Determinant:	1.2033.6t1.a.b
Projective image:	$A_4$
Projective stem field:	Galois closure of 4.0.4133089.1

Defining polynomial

$f(x)$

$=$

$ x^{16} - 4 x^{15} + 10 x^{14} - 37 x^{13} + 93 x^{12} - 181 x^{11} + 370 x^{10} - 660 x^{9} + \cdots + 16321 $

The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$: $ x^{6} + 4x^{4} + 33x^{3} + 39x^{2} + 6x + 6 $ x^6 + 4*x^4 + 33*x^3 + 39*x^2 + 6*x + 6

Roots:

$r_{ 1 }$	$=$	$ 19 a^{5} + 20 a^{4} + 15 a^{3} + 16 a^{2} + 27 + \left(10 a^{5} + 28 a^{4} + 21 a^{3} + 30 a^{2} + 30 a + 32\right)\cdot 41 + \left(23 a^{5} + 19 a^{4} + 33 a^{3} + 19 a^{2} + 7 a + 32\right)\cdot 41^{2} + \left(9 a^{5} + 15 a^{4} + 27 a^{3} + 31 a^{2} + 33\right)\cdot 41^{3} + \left(12 a^{5} + 36 a^{4} + 40 a^{3} + 16 a^{2} + 22 a + 16\right)\cdot 41^{4} + \left(34 a^{5} + 40 a^{4} + 9 a^{2} + 8 a + 38\right)\cdot 41^{5} + \left(38 a^{5} + 9 a^{4} + 13 a^{3} + 5 a^{2} + 20 a + 22\right)\cdot 41^{6} + \left(21 a^{5} + 32 a^{4} + 24 a^{3} + 39 a^{2} + 20 a + 2\right)\cdot 41^{7} + \left(36 a^{5} + 34 a^{4} + 37 a^{3} + 5 a^{2} + 7 a + 22\right)\cdot 41^{8} + \left(34 a^{5} + a^{4} + 31 a^{3} + a^{2} + 23 a + 35\right)\cdot 41^{9} +O(41^{10})$ 19a^5 + 20a^4 + 15a^3 + 16a^2 + 27 + (10a^5 + 28a^4 + 21a^3 + 30a^2 + 30a + 32)41 + (23a^5 + 19a^4 + 33a^3 + 19a^2 + 7a + 32)41^2 + (9a^5 + 15a^4 + 27a^3 + 31a^2 + 33)41^3 + (12a^5 + 36a^4 + 40a^3 + 16a^2 + 22a + 16)41^4 + (34a^5 + 40a^4 + 9a^2 + 8a + 38)41^5 + (38a^5 + 9a^4 + 13a^3 + 5a^2 + 20a + 22)41^6 + (21a^5 + 32a^4 + 24a^3 + 39a^2 + 20a + 2)41^7 + (36a^5 + 34a^4 + 37a^3 + 5a^2 + 7a + 22)41^8 + (34a^5 + a^4 + 31a^3 + a^2 + 23a + 35)41^9+O(41^10)
$r_{ 2 }$	$=$	$ 33 a^{5} + 35 a^{4} + 6 a^{3} + 40 a^{2} + 6 a + 36 + \left(34 a^{5} + 11 a^{4} + 33 a^{3} + 23 a^{2} + 5 a + 28\right)\cdot 41 + \left(27 a^{5} + 5 a^{4} + 35 a^{3} + 39 a^{2} + 5 a + 21\right)\cdot 41^{2} + \left(13 a^{5} + 39 a^{4} + 10 a^{3} + 35 a^{2} + 18 a + 37\right)\cdot 41^{3} + \left(10 a^{5} + 27 a^{4} + 5 a^{3} + 13 a^{2} + 8 a + 27\right)\cdot 41^{4} + \left(a^{5} + 7 a^{4} + 33 a^{3} + 34 a^{2} + 31 a + 27\right)\cdot 41^{5} + \left(37 a^{5} + 13 a^{4} + 30 a^{3} + 25 a^{2} + 27 a + 13\right)\cdot 41^{6} + \left(38 a^{5} + 32 a^{4} + 9 a^{3} + 15 a^{2} + 8 a + 35\right)\cdot 41^{7} + \left(39 a^{5} + 26 a^{4} + 5 a^{3} + 25 a^{2} + 17 a + 11\right)\cdot 41^{8} + \left(3 a^{5} + 2 a^{4} + 30 a^{3} + 14 a^{2} + 24 a + 5\right)\cdot 41^{9} +O(41^{10})$ 33a^5 + 35a^4 + 6a^3 + 40a^2 + 6a + 36 + (34a^5 + 11a^4 + 33a^3 + 23a^2 + 5a + 28)41 + (27a^5 + 5a^4 + 35a^3 + 39a^2 + 5a + 21)41^2 + (13a^5 + 39a^4 + 10a^3 + 35a^2 + 18a + 37)41^3 + (10a^5 + 27a^4 + 5a^3 + 13a^2 + 8a + 27)41^4 + (a^5 + 7a^4 + 33a^3 + 34a^2 + 31a + 27)41^5 + (37a^5 + 13a^4 + 30a^3 + 25a^2 + 27a + 13)41^6 + (38a^5 + 32a^4 + 9a^3 + 15a^2 + 8a + 35)41^7 + (39a^5 + 26a^4 + 5a^3 + 25a^2 + 17a + 11)41^8 + (3a^5 + 2a^4 + 30a^3 + 14a^2 + 24a + 5)41^9+O(41^10)
$r_{ 3 }$	$=$	$ 25 a^{5} + 9 a^{4} + 35 a^{3} + 22 a^{2} + 31 a + 13 + \left(19 a^{5} + 37 a^{4} + 5 a^{3} + 40 a^{2} + 40 a + 27\right)\cdot 41 + \left(30 a^{5} + 18 a^{4} + 21 a^{3} + 29 a^{2} + 27 a + 21\right)\cdot 41^{2} + \left(34 a^{5} + a^{4} + 14 a^{3} + 3 a^{2} + 18 a + 9\right)\cdot 41^{3} + \left(37 a^{5} + 5 a^{4} + 38 a^{3} + 19 a^{2} + 33 a\right)\cdot 41^{4} + \left(30 a^{5} + 13 a^{4} + 3 a^{3} + 9 a^{2} + 24 a + 2\right)\cdot 41^{5} + \left(35 a^{5} + 8 a^{4} + 20 a^{3} + 34 a^{2} + 27 a + 33\right)\cdot 41^{6} + \left(12 a^{5} + 12 a^{4} + 16 a^{3} + 16 a^{2} + 13 a + 40\right)\cdot 41^{7} + \left(5 a^{5} + 18 a^{4} + 24 a^{3} + 6 a^{2} + 18 a + 22\right)\cdot 41^{8} + \left(29 a^{5} + 38 a^{4} + 25 a^{3} + 5 a^{2} + 18 a + 4\right)\cdot 41^{9} +O(41^{10})$ 25a^5 + 9a^4 + 35a^3 + 22a^2 + 31a + 13 + (19a^5 + 37a^4 + 5a^3 + 40a^2 + 40a + 27)41 + (30a^5 + 18a^4 + 21a^3 + 29a^2 + 27a + 21)41^2 + (34a^5 + a^4 + 14a^3 + 3a^2 + 18a + 9)41^3 + (37a^5 + 5a^4 + 38a^3 + 19a^2 + 33a)41^4 + (30a^5 + 13a^4 + 3a^3 + 9a^2 + 24a + 2)41^5 + (35a^5 + 8a^4 + 20a^3 + 34a^2 + 27a + 33)41^6 + (12a^5 + 12a^4 + 16a^3 + 16a^2 + 13a + 40)41^7 + (5a^5 + 18a^4 + 24a^3 + 6a^2 + 18a + 22)41^8 + (29a^5 + 38a^4 + 25a^3 + 5a^2 + 18a + 4)41^9+O(41^10)
$r_{ 4 }$	$=$	$ 7 a^{5} + 11 a^{4} + 38 a^{3} + 36 a^{2} + 5 a + 32 + \left(19 a^{4} + 12 a^{3} + 28 a^{2} + 21 a + 40\right)\cdot 41 + \left(24 a^{5} + 9 a^{4} + 3 a^{3} + 9 a^{2} + 4 a + 10\right)\cdot 41^{2} + \left(12 a^{5} + 13 a^{4} + 20 a^{3} + 13 a^{2} + 4 a + 5\right)\cdot 41^{3} + \left(3 a^{5} + 23 a^{4} + 8 a^{3} + 39 a^{2} + 19 a + 19\right)\cdot 41^{4} + \left(11 a^{5} + 3 a^{4} + 35 a^{3} + 3 a^{2} + 33 a + 11\right)\cdot 41^{5} + \left(33 a^{5} + 18 a^{4} + 29 a^{3} + 37 a^{2} + 9 a + 10\right)\cdot 41^{6} + \left(8 a^{5} + 24 a^{4} + 11 a^{3} + 28 a^{2} + 27 a + 14\right)\cdot 41^{7} + \left(15 a^{5} + 28 a^{4} + 6 a^{3} + 15 a^{2} + 2 a + 40\right)\cdot 41^{8} + \left(11 a^{5} + 6 a^{4} + 34 a^{3} + 28 a^{2} + 34 a + 11\right)\cdot 41^{9} +O(41^{10})$ 7a^5 + 11a^4 + 38a^3 + 36a^2 + 5a + 32 + (19a^4 + 12a^3 + 28a^2 + 21a + 40)41 + (24a^5 + 9a^4 + 3a^3 + 9a^2 + 4a + 10)41^2 + (12a^5 + 13a^4 + 20a^3 + 13a^2 + 4a + 5)41^3 + (3a^5 + 23a^4 + 8a^3 + 39a^2 + 19a + 19)41^4 + (11a^5 + 3a^4 + 35a^3 + 3a^2 + 33a + 11)41^5 + (33a^5 + 18a^4 + 29a^3 + 37a^2 + 9a + 10)41^6 + (8a^5 + 24a^4 + 11a^3 + 28a^2 + 27a + 14)41^7 + (15a^5 + 28a^4 + 6a^3 + 15a^2 + 2a + 40)41^8 + (11a^5 + 6a^4 + 34a^3 + 28a^2 + 34a + 11)41^9+O(41^10)
$r_{ 5 }$	$=$	$ 23 a^{4} + 39 a^{3} + 5 a^{2} + 28 a + 5 + \left(a^{5} + 16 a^{4} + 2 a^{3} + 38 a^{2} + 25 a + 34\right)\cdot 41 + \left(10 a^{5} + 16 a^{4} + 31 a^{3} + 9 a^{2} + 30 a + 33\right)\cdot 41^{2} + \left(7 a^{5} + 9 a^{4} + 22 a^{3} + 18 a^{2} + 12 a + 38\right)\cdot 41^{3} + \left(29 a^{5} + 17 a^{4} + 19 a^{3} + 12 a^{2} + 35 a + 9\right)\cdot 41^{4} + \left(31 a^{5} + a^{4} + 23 a^{3} + 20 a^{2} + 29 a + 14\right)\cdot 41^{5} + \left(10 a^{5} + 25 a^{4} + 39 a^{3} + a^{2} + 15 a + 33\right)\cdot 41^{6} + \left(33 a^{5} + 7 a^{4} + 9 a^{2} + 25 a + 18\right)\cdot 41^{7} + \left(3 a^{5} + 32 a^{4} + 33 a^{3} + 4 a^{2} + 13 a + 19\right)\cdot 41^{8} + \left(31 a^{5} + 15 a^{4} + 20 a^{3} + a^{2} + 31 a + 31\right)\cdot 41^{9} +O(41^{10})$ 23a^4 + 39a^3 + 5a^2 + 28a + 5 + (a^5 + 16a^4 + 2a^3 + 38a^2 + 25a + 34)41 + (10a^5 + 16a^4 + 31a^3 + 9a^2 + 30a + 33)41^2 + (7a^5 + 9a^4 + 22a^3 + 18a^2 + 12a + 38)41^3 + (29a^5 + 17a^4 + 19a^3 + 12a^2 + 35a + 9)41^4 + (31a^5 + a^4 + 23a^3 + 20a^2 + 29a + 14)41^5 + (10a^5 + 25a^4 + 39a^3 + a^2 + 15a + 33)41^6 + (33a^5 + 7a^4 + 9a^2 + 25a + 18)41^7 + (3a^5 + 32a^4 + 33a^3 + 4a^2 + 13a + 19)41^8 + (31a^5 + 15a^4 + 20a^3 + a^2 + 31a + 31)*41^9+O(41^10)
$r_{ 6 }$	$=$	$ 3 a^{5} + a^{4} + 11 a^{3} + 9 a^{2} + 26 + \left(13 a^{5} + 9 a^{4} + 27 a^{3} + 22 a + 26\right)\cdot 41 + \left(10 a^{5} + 14 a^{4} + 34 a^{3} + 17 a^{2} + a + 18\right)\cdot 41^{2} + \left(27 a^{5} + 25 a^{4} + 22 a^{3} + 39 a^{2} + 26 a + 17\right)\cdot 41^{3} + \left(8 a^{5} + 5 a^{4} + 14 a^{3} + 17 a^{2} + 28 a + 5\right)\cdot 41^{4} + \left(19 a^{5} + 4 a^{4} + 8 a^{3} + 39 a^{2} + 30 a + 21\right)\cdot 41^{5} + \left(3 a^{5} + 2 a^{4} + 6 a^{3} + 6 a + 21\right)\cdot 41^{6} + \left(6 a^{5} + 29 a^{4} + 4 a^{3} + 15 a^{2} + 6 a + 3\right)\cdot 41^{7} + \left(22 a^{5} + 30 a^{4} + 16 a^{3} + 30 a^{2} + 9 a + 3\right)\cdot 41^{8} + \left(15 a^{5} + 39 a^{4} + 3 a^{3} + 22 a^{2} + 7 a + 30\right)\cdot 41^{9} +O(41^{10})$ 3a^5 + a^4 + 11a^3 + 9a^2 + 26 + (13a^5 + 9a^4 + 27a^3 + 22a + 26)41 + (10a^5 + 14a^4 + 34a^3 + 17a^2 + a + 18)41^2 + (27a^5 + 25a^4 + 22a^3 + 39a^2 + 26a + 17)41^3 + (8a^5 + 5a^4 + 14a^3 + 17a^2 + 28a + 5)41^4 + (19a^5 + 4a^4 + 8a^3 + 39a^2 + 30a + 21)41^5 + (3a^5 + 2a^4 + 6a^3 + 6a + 21)41^6 + (6a^5 + 29a^4 + 4a^3 + 15a^2 + 6a + 3)41^7 + (22a^5 + 30a^4 + 16a^3 + 30a^2 + 9a + 3)41^8 + (15a^5 + 39a^4 + 3a^3 + 22a^2 + 7a + 30)41^9+O(41^10)
$r_{ 7 }$	$=$	$ 26 a^{5} + 28 a^{4} + 11 a^{3} + 30 a^{2} + 19 a + 19 + \left(8 a^{5} + 5 a^{4} + 7 a^{3} + 13 a^{2} + 26 a + 1\right)\cdot 41 + \left(3 a^{5} + 40 a^{4} + 29 a^{3} + 32 a^{2} + 4 a + 5\right)\cdot 41^{2} + \left(26 a^{5} + 39 a^{4} + 10 a^{3} + 13 a^{2} + 33 a + 26\right)\cdot 41^{3} + \left(22 a^{5} + 4 a^{4} + 3 a^{3} + 19 a^{2} + 9 a + 32\right)\cdot 41^{4} + \left(11 a^{5} + 40 a^{4} + 28 a^{3} + 11 a^{2} + 28 a + 35\right)\cdot 41^{5} + \left(29 a^{5} + 7 a^{4} + 28 a^{3} + 19 a^{2} + 14 a + 19\right)\cdot 41^{6} + \left(5 a^{5} + 29 a^{4} + 37 a^{2} + 15 a + 7\right)\cdot 41^{7} + \left(40 a^{5} + 29 a^{3} + 6 a^{2} + 19 a + 5\right)\cdot 41^{8} + \left(18 a^{5} + 38 a^{4} + 10 a^{3} + 19 a^{2} + 13 a + 1\right)\cdot 41^{9} +O(41^{10})$ 26a^5 + 28a^4 + 11a^3 + 30a^2 + 19a + 19 + (8a^5 + 5a^4 + 7a^3 + 13a^2 + 26a + 1)41 + (3a^5 + 40a^4 + 29a^3 + 32a^2 + 4a + 5)41^2 + (26a^5 + 39a^4 + 10a^3 + 13a^2 + 33a + 26)41^3 + (22a^5 + 4a^4 + 3a^3 + 19a^2 + 9a + 32)41^4 + (11a^5 + 40a^4 + 28a^3 + 11a^2 + 28a + 35)41^5 + (29a^5 + 7a^4 + 28a^3 + 19a^2 + 14a + 19)41^6 + (5a^5 + 29a^4 + 37a^2 + 15a + 7)41^7 + (40a^5 + 29a^3 + 6a^2 + 19a + 5)41^8 + (18a^5 + 38a^4 + 10a^3 + 19a^2 + 13a + 1)*41^9+O(41^10)
$r_{ 8 }$	$=$	$ 7 a^{5} + 36 a^{4} + 24 a^{3} + 17 a^{2} + 10 a + 11 + \left(17 a^{5} + 8 a^{4} + 8 a^{3} + 7 a^{2} + 30 a + 40\right)\cdot 41 + \left(6 a^{5} + 27 a^{4} + 28 a^{3} + 16 a^{2} + 4 a + 22\right)\cdot 41^{2} + \left(36 a^{5} + 3 a^{4} + a^{3} + 22 a^{2} + 31 a + 20\right)\cdot 41^{3} + \left(35 a^{5} + 28 a^{4} + a^{3} + 26 a^{2} + 27 a + 29\right)\cdot 41^{4} + \left(20 a^{5} + 24 a^{4} + 39 a^{3} + 36 a^{2} + 35 a + 4\right)\cdot 41^{5} + \left(24 a^{5} + 34 a^{4} + 14 a^{3} + 36 a^{2} + 7 a + 15\right)\cdot 41^{6} + \left(36 a^{5} + 16 a^{4} + 5 a^{3} + 36 a^{2} + 4 a + 3\right)\cdot 41^{7} + \left(37 a^{5} + 15 a^{4} + 38 a^{3} + 14 a^{2} + a + 33\right)\cdot 41^{8} + \left(35 a^{5} + 29 a^{4} + 22 a^{3} + 9 a^{2} + 5 a + 28\right)\cdot 41^{9} +O(41^{10})$ 7a^5 + 36a^4 + 24a^3 + 17a^2 + 10a + 11 + (17a^5 + 8a^4 + 8a^3 + 7a^2 + 30a + 40)41 + (6a^5 + 27a^4 + 28a^3 + 16a^2 + 4a + 22)41^2 + (36a^5 + 3a^4 + a^3 + 22a^2 + 31a + 20)41^3 + (35a^5 + 28a^4 + a^3 + 26a^2 + 27a + 29)41^4 + (20a^5 + 24a^4 + 39a^3 + 36a^2 + 35a + 4)41^5 + (24a^5 + 34a^4 + 14a^3 + 36a^2 + 7a + 15)41^6 + (36a^5 + 16a^4 + 5a^3 + 36a^2 + 4a + 3)41^7 + (37a^5 + 15a^4 + 38a^3 + 14a^2 + a + 33)41^8 + (35a^5 + 29a^4 + 22a^3 + 9a^2 + 5a + 28)41^9+O(41^10)
$r_{ 9 }$	$=$	$ 22 a^{5} + 21 a^{4} + 26 a^{3} + 25 a^{2} + 15 + \left(30 a^{5} + 12 a^{4} + 19 a^{3} + 10 a^{2} + 11 a + 20\right)\cdot 41 + \left(17 a^{5} + 21 a^{4} + 7 a^{3} + 21 a^{2} + 33 a + 32\right)\cdot 41^{2} + \left(31 a^{5} + 25 a^{4} + 13 a^{3} + 9 a^{2} + 40 a + 4\right)\cdot 41^{3} + \left(28 a^{5} + 4 a^{4} + 24 a^{2} + 18 a + 23\right)\cdot 41^{4} + \left(6 a^{5} + 40 a^{3} + 31 a^{2} + 32 a + 26\right)\cdot 41^{5} + \left(2 a^{5} + 31 a^{4} + 27 a^{3} + 35 a^{2} + 20 a + 23\right)\cdot 41^{6} + \left(19 a^{5} + 8 a^{4} + 16 a^{3} + a^{2} + 20 a + 7\right)\cdot 41^{7} + \left(4 a^{5} + 6 a^{4} + 3 a^{3} + 35 a^{2} + 33 a + 39\right)\cdot 41^{8} + \left(6 a^{5} + 39 a^{4} + 9 a^{3} + 39 a^{2} + 17 a + 38\right)\cdot 41^{9} +O(41^{10})$ 22a^5 + 21a^4 + 26a^3 + 25a^2 + 15 + (30a^5 + 12a^4 + 19a^3 + 10a^2 + 11a + 20)41 + (17a^5 + 21a^4 + 7a^3 + 21a^2 + 33a + 32)41^2 + (31a^5 + 25a^4 + 13a^3 + 9a^2 + 40a + 4)41^3 + (28a^5 + 4a^4 + 24a^2 + 18a + 23)41^4 + (6a^5 + 40a^3 + 31a^2 + 32a + 26)41^5 + (2a^5 + 31a^4 + 27a^3 + 35a^2 + 20a + 23)41^6 + (19a^5 + 8a^4 + 16a^3 + a^2 + 20a + 7)41^7 + (4a^5 + 6a^4 + 3a^3 + 35a^2 + 33a + 39)41^8 + (6a^5 + 39a^4 + 9a^3 + 39a^2 + 17a + 38)*41^9+O(41^10)
$r_{ 10 }$	$=$	$ 5 a^{5} + 36 a^{4} + 2 a^{3} + 31 a^{2} + 35 + \left(3 a^{5} + 31 a^{4} + 23 a^{3} + 26 a^{2} + 21 a + 25\right)\cdot 41 + \left(8 a^{5} + 32 a^{4} + 37 a^{3} + 36 a^{2} + 33 a + 39\right)\cdot 41^{2} + \left(8 a^{5} + 10 a^{4} + 16 a^{3} + 5 a^{2} + 8 a + 18\right)\cdot 41^{3} + \left(6 a^{5} + 27 a^{4} + 19 a^{3} + 40 a^{2} + 15 a + 5\right)\cdot 41^{4} + \left(35 a^{5} + 40 a^{4} + 10 a^{3} + 15 a^{2} + 23 a + 11\right)\cdot 41^{5} + \left(6 a^{5} + 8 a^{4} + 6 a^{3} + 15 a^{2} + 10 a + 4\right)\cdot 41^{6} + \left(34 a^{5} + 31 a^{4} + 20 a^{3} + 15 a^{2} + 37 a + 25\right)\cdot 41^{7} + \left(20 a^{5} + 24 a^{4} + 23 a^{3} + a^{2} + 32 a + 18\right)\cdot 41^{8} + \left(35 a^{5} + 40 a^{4} + 34 a^{3} + 29 a^{2} + 37 a + 17\right)\cdot 41^{9} +O(41^{10})$ 5a^5 + 36a^4 + 2a^3 + 31a^2 + 35 + (3a^5 + 31a^4 + 23a^3 + 26a^2 + 21a + 25)41 + (8a^5 + 32a^4 + 37a^3 + 36a^2 + 33a + 39)41^2 + (8a^5 + 10a^4 + 16a^3 + 5a^2 + 8a + 18)41^3 + (6a^5 + 27a^4 + 19a^3 + 40a^2 + 15a + 5)41^4 + (35a^5 + 40a^4 + 10a^3 + 15a^2 + 23a + 11)41^5 + (6a^5 + 8a^4 + 6a^3 + 15a^2 + 10a + 4)41^6 + (34a^5 + 31a^4 + 20a^3 + 15a^2 + 37a + 25)41^7 + (20a^5 + 24a^4 + 23a^3 + a^2 + 32a + 18)41^8 + (35a^5 + 40a^4 + 34a^3 + 29a^2 + 37a + 17)*41^9+O(41^10)
$r_{ 11 }$	$=$	$ 2 a^{5} + 26 a^{4} + 11 a^{3} + 5 a^{2} + 3 a + 10 + \left(15 a^{5} + 40 a^{4} + 7 a^{3} + 19 a^{2} + 10 a + 11\right)\cdot 41 + \left(6 a^{5} + 27 a^{4} + 15 a^{3} + 37 a^{2} + 7 a + 1\right)\cdot 41^{2} + \left(8 a^{5} + 4 a^{4} + 34 a^{3} + 26 a^{2} + 40 a + 1\right)\cdot 41^{3} + \left(a^{5} + 35 a^{4} + 32 a^{3} + 9 a^{2} + 17 a + 34\right)\cdot 41^{4} + \left(21 a^{5} + 21 a^{4} + 35 a^{3} + 34 a^{2} + 8 a + 30\right)\cdot 41^{5} + \left(5 a^{5} + 13 a^{4} + 27 a^{3} + 28 a^{2} + 11 a + 13\right)\cdot 41^{6} + \left(33 a^{5} + 26 a^{4} + 26 a^{3} + 30 a^{2} + 16 a + 11\right)\cdot 41^{7} + \left(5 a^{5} + 19 a^{4} + 21 a^{3} + 14 a^{2} + 37 a + 19\right)\cdot 41^{8} + \left(28 a^{5} + 35 a^{4} + 4 a^{3} + 38 a^{2} + 30 a + 37\right)\cdot 41^{9} +O(41^{10})$ 2a^5 + 26a^4 + 11a^3 + 5a^2 + 3a + 10 + (15a^5 + 40a^4 + 7a^3 + 19a^2 + 10a + 11)41 + (6a^5 + 27a^4 + 15a^3 + 37a^2 + 7a + 1)41^2 + (8a^5 + 4a^4 + 34a^3 + 26a^2 + 40a + 1)41^3 + (a^5 + 35a^4 + 32a^3 + 9a^2 + 17a + 34)41^4 + (21a^5 + 21a^4 + 35a^3 + 34a^2 + 8a + 30)41^5 + (5a^5 + 13a^4 + 27a^3 + 28a^2 + 11a + 13)41^6 + (33a^5 + 26a^4 + 26a^3 + 30a^2 + 16a + 11)41^7 + (5a^5 + 19a^4 + 21a^3 + 14a^2 + 37a + 19)41^8 + (28a^5 + 35a^4 + 4a^3 + 38a^2 + 30a + 37)41^9+O(41^10)
$r_{ 12 }$	$=$	$ 25 a^{5} + 6 a^{4} + 4 a^{3} + 18 a^{2} + 16 a + 23 + \left(22 a^{5} + 12 a^{4} + 33 a^{3} + 11 a^{2} + 10 a + 40\right)\cdot 41 + \left(25 a^{5} + 27 a^{4} + 15 a^{3} + 40 a^{2} + 6 a + 5\right)\cdot 41^{2} + \left(32 a^{5} + 19 a^{4} + 35 a^{3} + 13 a^{2} + 16 a + 37\right)\cdot 41^{3} + \left(19 a^{5} + 7 a^{4} + 40 a^{3} + 2 a^{2} + 16 a + 16\right)\cdot 41^{4} + \left(4 a^{5} + 33 a^{4} + 32 a^{3} + 7 a^{2} + 33\right)\cdot 41^{5} + \left(11 a^{5} + 15 a^{4} + 28 a^{3} + 30 a^{2} + 17 a + 35\right)\cdot 41^{6} + \left(3 a^{5} + 5 a^{4} + 38 a^{3} + 6 a^{2} + 21 a + 36\right)\cdot 41^{7} + \left(40 a^{5} + 40 a^{4} + 30 a^{3} + 9 a^{2} + 8 a + 40\right)\cdot 41^{8} + \left(39 a^{5} + 11 a^{4} + 31 a^{3} + 21 a^{2} + 33\right)\cdot 41^{9} +O(41^{10})$ 25a^5 + 6a^4 + 4a^3 + 18a^2 + 16a + 23 + (22a^5 + 12a^4 + 33a^3 + 11a^2 + 10a + 40)41 + (25a^5 + 27a^4 + 15a^3 + 40a^2 + 6a + 5)41^2 + (32a^5 + 19a^4 + 35a^3 + 13a^2 + 16a + 37)41^3 + (19a^5 + 7a^4 + 40a^3 + 2a^2 + 16a + 16)41^4 + (4a^5 + 33a^4 + 32a^3 + 7a^2 + 33)41^5 + (11a^5 + 15a^4 + 28a^3 + 30a^2 + 17a + 35)41^6 + (3a^5 + 5a^4 + 38a^3 + 6a^2 + 21a + 36)41^7 + (40a^5 + 40a^4 + 30a^3 + 9a^2 + 8a + 40)41^8 + (39a^5 + 11a^4 + 31a^3 + 21a^2 + 33)*41^9+O(41^10)
$r_{ 13 }$	$=$	$ 36 a^{5} + 38 a^{4} + 10 a^{3} + a^{2} + 13 a + 35 + \left(34 a^{5} + a^{4} + 25 a^{3} + 37 a^{2} + 5 a + 25\right)\cdot 41 + \left(13 a^{5} + 5 a^{4} + 18 a^{3} + 39 a^{2} + 38 a + 17\right)\cdot 41^{2} + \left(25 a^{5} + 16 a^{4} + 24 a^{3} + 35 a^{2} + 15 a + 17\right)\cdot 41^{3} + \left(37 a^{5} + 25 a^{4} + 9 a^{3} + 35 a^{2} + 31 a + 13\right)\cdot 41^{4} + \left(32 a^{5} + 5 a^{4} + 37 a^{3} + 22 a^{2} + 13 a + 20\right)\cdot 41^{5} + \left(8 a^{5} + 28 a^{4} + 27 a^{3} + 33 a^{2} + 25 a + 38\right)\cdot 41^{6} + \left(27 a^{5} + 27 a^{4} + 10 a^{3} + 40 a^{2} + 35 a + 20\right)\cdot 41^{7} + \left(7 a^{5} + 38 a^{4} + 21 a^{3} + 24 a^{2} + 34 a + 40\right)\cdot 41^{8} + \left(9 a^{5} + 4 a^{4} + 40 a^{3} + 22 a^{2} + 12 a + 21\right)\cdot 41^{9} +O(41^{10})$ 36a^5 + 38a^4 + 10a^3 + a^2 + 13a + 35 + (34a^5 + a^4 + 25a^3 + 37a^2 + 5a + 25)41 + (13a^5 + 5a^4 + 18a^3 + 39a^2 + 38a + 17)41^2 + (25a^5 + 16a^4 + 24a^3 + 35a^2 + 15a + 17)41^3 + (37a^5 + 25a^4 + 9a^3 + 35a^2 + 31a + 13)41^4 + (32a^5 + 5a^4 + 37a^3 + 22a^2 + 13a + 20)41^5 + (8a^5 + 28a^4 + 27a^3 + 33a^2 + 25a + 38)41^6 + (27a^5 + 27a^4 + 10a^3 + 40a^2 + 35a + 20)41^7 + (7a^5 + 38a^4 + 21a^3 + 24a^2 + 34a + 40)41^8 + (9a^5 + 4a^4 + 40a^3 + 22a^2 + 12a + 21)*41^9+O(41^10)
$r_{ 14 }$	$=$	$ 38 a^{5} + 40 a^{4} + 30 a^{3} + 32 a^{2} + 9 + \left(27 a^{5} + 31 a^{4} + 13 a^{3} + 40 a^{2} + 19 a + 36\right)\cdot 41 + \left(30 a^{5} + 26 a^{4} + 6 a^{3} + 23 a^{2} + 39 a + 25\right)\cdot 41^{2} + \left(13 a^{5} + 15 a^{4} + 18 a^{3} + a^{2} + 14 a + 20\right)\cdot 41^{3} + \left(32 a^{5} + 35 a^{4} + 26 a^{3} + 23 a^{2} + 12 a + 39\right)\cdot 41^{4} + \left(21 a^{5} + 36 a^{4} + 32 a^{3} + a^{2} + 10 a + 4\right)\cdot 41^{5} + \left(37 a^{5} + 38 a^{4} + 34 a^{3} + 40 a^{2} + 34 a + 32\right)\cdot 41^{6} + \left(34 a^{5} + 11 a^{4} + 36 a^{3} + 25 a^{2} + 34 a + 12\right)\cdot 41^{7} + \left(18 a^{5} + 10 a^{4} + 24 a^{3} + 10 a^{2} + 31 a + 39\right)\cdot 41^{8} + \left(25 a^{5} + a^{4} + 37 a^{3} + 18 a^{2} + 33 a + 1\right)\cdot 41^{9} +O(41^{10})$ 38a^5 + 40a^4 + 30a^3 + 32a^2 + 9 + (27a^5 + 31a^4 + 13a^3 + 40a^2 + 19a + 36)41 + (30a^5 + 26a^4 + 6a^3 + 23a^2 + 39a + 25)41^2 + (13a^5 + 15a^4 + 18a^3 + a^2 + 14a + 20)41^3 + (32a^5 + 35a^4 + 26a^3 + 23a^2 + 12a + 39)41^4 + (21a^5 + 36a^4 + 32a^3 + a^2 + 10a + 4)41^5 + (37a^5 + 38a^4 + 34a^3 + 40a^2 + 34a + 32)41^6 + (34a^5 + 11a^4 + 36a^3 + 25a^2 + 34a + 12)41^7 + (18a^5 + 10a^4 + 24a^3 + 10a^2 + 31a + 39)41^8 + (25a^5 + a^4 + 37a^3 + 18a^2 + 33a + 1)*41^9+O(41^10)
$r_{ 15 }$	$=$	$ 27 a^{5} + 7 a^{4} + 21 a^{3} + 9 a^{2} + 36 a + 27 + \left(12 a^{5} + a^{4} + 13 a^{3} + 18 a^{2} + 38 a + 4\right)\cdot 41 + \left(34 a^{5} + 8 a^{4} + a^{3} + 5 a^{2} + 27 a + 36\right)\cdot 41^{2} + \left(29 a^{5} + 29 a^{3} + 40 a^{2} + a\right)\cdot 41^{3} + \left(19 a^{5} + 32 a^{4} + 4 a^{3} + 7 a^{2} + 13 a + 17\right)\cdot 41^{4} + \left(18 a^{5} + 38 a^{4} + 24 a^{3} + 9 a^{2} + 6 a + 18\right)\cdot 41^{5} + \left(5 a^{5} + 17 a^{4} + 39 a^{3} + 36 a^{2} + 2 a + 26\right)\cdot 41^{6} + \left(32 a^{5} + 18 a^{2} + 13 a + 28\right)\cdot 41^{7} + \left(7 a^{5} + 2 a^{4} + 28 a^{3} + 23 a^{2} + a + 4\right)\cdot 41^{8} + \left(13 a^{5} + 23 a^{4} + 22 a^{3} + 10 a^{2} + 13 a + 21\right)\cdot 41^{9} +O(41^{10})$ 27a^5 + 7a^4 + 21a^3 + 9a^2 + 36a + 27 + (12a^5 + a^4 + 13a^3 + 18a^2 + 38a + 4)41 + (34a^5 + 8a^4 + a^3 + 5a^2 + 27a + 36)41^2 + (29a^5 + 29a^3 + 40a^2 + a)41^3 + (19a^5 + 32a^4 + 4a^3 + 7a^2 + 13a + 17)41^4 + (18a^5 + 38a^4 + 24a^3 + 9a^2 + 6a + 18)41^5 + (5a^5 + 17a^4 + 39a^3 + 36a^2 + 2a + 26)41^6 + (32a^5 + 18a^2 + 13a + 28)41^7 + (7a^5 + 2a^4 + 28a^3 + 23a^2 + a + 4)41^8 + (13a^5 + 23a^4 + 22a^3 + 10a^2 + 13a + 21)41^9+O(41^10)
$r_{ 16 }$	$=$	$ 12 a^{5} + 32 a^{4} + 4 a^{3} + 32 a^{2} + 38 a + 9 + \left(35 a^{5} + 17 a^{4} + 32 a^{3} + 21 a^{2} + 10 a + 13\right)\cdot 41 + \left(14 a^{5} + 27 a^{4} + 8 a^{3} + 30 a^{2} + 14 a + 1\right)\cdot 41^{2} + \left(11 a^{5} + 5 a^{4} + 25 a^{3} + 15 a^{2} + 4 a + 38\right)\cdot 41^{3} + \left(22 a^{5} + 12 a^{4} + 21 a^{3} + 19 a^{2} + 18 a + 36\right)\cdot 41^{4} + \left(26 a^{5} + 15 a^{4} + 24 a^{3} + 40 a^{2} + 10 a + 26\right)\cdot 41^{5} + \left(37 a^{5} + 13 a^{4} + 33 a^{3} + 28 a^{2} + 35 a + 24\right)\cdot 41^{6} + \left(20 a^{5} + 32 a^{4} + 21 a^{3} + 29 a^{2} + 27 a + 17\right)\cdot 41^{7} + \left(21 a^{5} + 39 a^{4} + 25 a^{3} + 16 a^{2} + 17 a + 8\right)\cdot 41^{8} + \left(30 a^{5} + 39 a^{4} + 8 a^{3} + 5 a^{2} + 24 a + 6\right)\cdot 41^{9} +O(41^{10})$ 12a^5 + 32a^4 + 4a^3 + 32a^2 + 38a + 9 + (35a^5 + 17a^4 + 32a^3 + 21a^2 + 10a + 13)41 + (14a^5 + 27a^4 + 8a^3 + 30a^2 + 14a + 1)41^2 + (11a^5 + 5a^4 + 25a^3 + 15a^2 + 4a + 38)41^3 + (22a^5 + 12a^4 + 21a^3 + 19a^2 + 18a + 36)41^4 + (26a^5 + 15a^4 + 24a^3 + 40a^2 + 10a + 26)41^5 + (37a^5 + 13a^4 + 33a^3 + 28a^2 + 35a + 24)41^6 + (20a^5 + 32a^4 + 21a^3 + 29a^2 + 27a + 17)41^7 + (21a^5 + 39a^4 + 25a^3 + 16a^2 + 17a + 8)41^8 + (30a^5 + 39a^4 + 8a^3 + 5a^2 + 24a + 6)*41^9+O(41^10)

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.

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Elliptic curves over $\Q$
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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	2.2033.16t60.a.a
Dimension	$2$
Group	$\SL(2,3):C_2$
Conductor	$2033$
Root number	not computed
Indicator	$0$