LMFDB - Artin representation 2.1184.16t60.a.a

Basic invariants

Dimension:	$2$
Group:	16T60
Conductor:	$1184$$\medspace = 2^{5} \cdot 37 $
Artin stem field:	Galois closure of 16.0.60343937529850535870464.5
Galois orbit size:	$4$
Smallest permutation container:	16T60
Parity:	odd
Determinant:	1.148.6t1.b.b
Projective image:	$A_4$
Projective stem field:	Galois closure of 4.0.87616.1

Defining polynomial

$f(x)$

$=$

$ x^{16} - 8x^{14} - 12x^{12} + 184x^{10} + 486x^{8} - 184x^{6} - 12x^{4} + 8x^{2} + 1 $

x^16 - 8*x^14 - 12*x^12 + 184*x^10 + 486*x^8 - 184*x^6 - 12*x^4 + 8*x^2 + 1

The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{6} + x^{4} + 7x^{3} + 4x^{2} + 45x + 2 $ x^6 + x^4 + 7*x^3 + 4*x^2 + 45*x + 2

Roots:

$r_{ 1 }$	$=$	$ 15 a^{5} + a^{4} + 8 a^{3} + 33 a^{2} + 9 a + 4 + \left(43 a^{5} + 18 a^{4} + 13 a^{3} + 37 a^{2} + 9 a + 49\right)\cdot 53 + \left(51 a^{5} + 32 a^{3} + 32 a^{2} + 10 a + 42\right)\cdot 53^{2} + \left(38 a^{5} + 22 a^{4} + 19 a^{3} + 23 a^{2} + 22 a + 1\right)\cdot 53^{3} + \left(46 a^{5} + 13 a^{4} + 38 a^{3} + 7 a^{2} + 3 a + 5\right)\cdot 53^{4} + \left(36 a^{5} + 8 a^{4} + 33 a^{3} + 4 a^{2} + 10 a + 8\right)\cdot 53^{5} + \left(33 a^{5} + 46 a^{4} + 27 a^{3} + 23 a^{2} + 4 a + 7\right)\cdot 53^{6} + \left(13 a^{5} + 9 a^{4} + 5 a^{3} + 19 a^{2} + 24 a + 30\right)\cdot 53^{7} + \left(17 a^{5} + 15 a^{4} + 10 a^{3} + 49 a^{2} + 7 a + 6\right)\cdot 53^{8} + \left(2 a^{5} + 11 a^{4} + 20 a^{3} + 22 a + 3\right)\cdot 53^{9} +O(53^{10})$ 15a^5 + a^4 + 8a^3 + 33a^2 + 9a + 4 + (43a^5 + 18a^4 + 13a^3 + 37a^2 + 9a + 49)53 + (51a^5 + 32a^3 + 32a^2 + 10a + 42)53^2 + (38a^5 + 22a^4 + 19a^3 + 23a^2 + 22a + 1)53^3 + (46a^5 + 13a^4 + 38a^3 + 7a^2 + 3a + 5)53^4 + (36a^5 + 8a^4 + 33a^3 + 4a^2 + 10a + 8)53^5 + (33a^5 + 46a^4 + 27a^3 + 23a^2 + 4a + 7)53^6 + (13a^5 + 9a^4 + 5a^3 + 19a^2 + 24a + 30)53^7 + (17a^5 + 15a^4 + 10a^3 + 49a^2 + 7a + 6)53^8 + (2a^5 + 11a^4 + 20a^3 + 22a + 3)53^9+O(53^10)
$r_{ 2 }$	$=$	$ 36 a^{5} + 46 a^{4} + 52 a^{3} + 46 a^{2} + 46 a + 49 + \left(9 a^{5} + 50 a^{4} + 8 a^{3} + 23 a^{2} + 21 a + 22\right)\cdot 53 + \left(27 a^{5} + 28 a^{4} + 32 a^{3} + 42 a^{2} + 3 a + 30\right)\cdot 53^{2} + \left(4 a^{5} + 45 a^{3} + 47 a^{2} + 39 a + 10\right)\cdot 53^{3} + \left(51 a^{5} + 50 a^{4} + 43 a^{3} + 4 a^{2} + 6 a + 52\right)\cdot 53^{4} + \left(20 a^{5} + 25 a^{4} + 52 a^{3} + a^{2} + 22 a + 43\right)\cdot 53^{5} + \left(15 a^{5} + 25 a^{4} + 30 a^{3} + 49 a^{2} + 14 a + 35\right)\cdot 53^{6} + \left(31 a^{5} + 3 a^{4} + 44 a^{3} + 15 a^{2} + 17 a + 29\right)\cdot 53^{7} + \left(23 a^{5} + 48 a^{4} + 21 a^{3} + 48 a^{2} + 21 a + 6\right)\cdot 53^{8} + \left(47 a^{5} + 27 a^{4} + 14 a^{3} + 49 a^{2} + 5 a + 18\right)\cdot 53^{9} +O(53^{10})$ 36a^5 + 46a^4 + 52a^3 + 46a^2 + 46a + 49 + (9a^5 + 50a^4 + 8a^3 + 23a^2 + 21a + 22)53 + (27a^5 + 28a^4 + 32a^3 + 42a^2 + 3a + 30)53^2 + (4a^5 + 45a^3 + 47a^2 + 39a + 10)53^3 + (51a^5 + 50a^4 + 43a^3 + 4a^2 + 6a + 52)53^4 + (20a^5 + 25a^4 + 52a^3 + a^2 + 22a + 43)53^5 + (15a^5 + 25a^4 + 30a^3 + 49a^2 + 14a + 35)53^6 + (31a^5 + 3a^4 + 44a^3 + 15a^2 + 17a + 29)53^7 + (23a^5 + 48a^4 + 21a^3 + 48a^2 + 21a + 6)53^8 + (47a^5 + 27a^4 + 14a^3 + 49a^2 + 5a + 18)*53^9+O(53^10)
$r_{ 3 }$	$=$	$ 23 a^{5} + 17 a^{4} + 18 a^{3} + 42 a^{2} + 24 a + 50 + \left(14 a^{5} + 38 a^{4} + 18 a^{3} + 49 a^{2} + 33 a + 8\right)\cdot 53 + \left(36 a^{5} + 29 a^{4} + 43 a^{3} + 7 a^{2} + 13 a + 3\right)\cdot 53^{2} + \left(50 a^{5} + 51 a^{4} + 19 a^{3} + 12 a^{2} + 13 a + 32\right)\cdot 53^{3} + \left(34 a^{5} + 14 a^{4} + 25 a^{3} + 21 a^{2} + 39 a + 27\right)\cdot 53^{4} + \left(3 a^{5} + 35 a^{4} + 16 a^{3} + 24 a^{2} + 38 a + 25\right)\cdot 53^{5} + \left(21 a^{5} + 3 a^{4} + 52 a^{3} + 10 a^{2} + 10 a + 5\right)\cdot 53^{6} + \left(50 a^{5} + 10 a^{4} + 36 a^{3} + 21 a^{2} + 8 a + 51\right)\cdot 53^{7} + \left(51 a^{5} + 24 a^{4} + 10 a^{3} + 29 a^{2} + 43\right)\cdot 53^{8} + \left(45 a^{5} + 28 a^{4} + 4 a^{3} + 30 a^{2} + 50 a + 36\right)\cdot 53^{9} +O(53^{10})$ 23a^5 + 17a^4 + 18a^3 + 42a^2 + 24a + 50 + (14a^5 + 38a^4 + 18a^3 + 49a^2 + 33a + 8)53 + (36a^5 + 29a^4 + 43a^3 + 7a^2 + 13a + 3)53^2 + (50a^5 + 51a^4 + 19a^3 + 12a^2 + 13a + 32)53^3 + (34a^5 + 14a^4 + 25a^3 + 21a^2 + 39a + 27)53^4 + (3a^5 + 35a^4 + 16a^3 + 24a^2 + 38a + 25)53^5 + (21a^5 + 3a^4 + 52a^3 + 10a^2 + 10a + 5)53^6 + (50a^5 + 10a^4 + 36a^3 + 21a^2 + 8a + 51)53^7 + (51a^5 + 24a^4 + 10a^3 + 29a^2 + 43)53^8 + (45a^5 + 28a^4 + 4a^3 + 30a^2 + 50a + 36)53^9+O(53^10)
$r_{ 4 }$	$=$	$ 27 a^{5} + 38 a^{4} + 24 a^{3} + 36 a^{2} + 4 a + 15 + \left(33 a^{5} + 51 a^{4} + 44 a^{3} + 41 a^{2} + 28 a + 26\right)\cdot 53 + \left(49 a^{5} + 3 a^{4} + 4 a^{3} + 15 a^{2} + 52 a + 48\right)\cdot 53^{2} + \left(13 a^{5} + 24 a^{4} + 42 a^{3} + 3 a^{2} + 36 a + 9\right)\cdot 53^{3} + \left(4 a^{5} + 6 a^{4} + 16 a^{3} + 44 a^{2} + 24 a + 29\right)\cdot 53^{4} + \left(18 a^{5} + 4 a^{4} + 47 a^{3} + 38 a^{2} + 3 a + 27\right)\cdot 53^{5} + \left(2 a^{5} + 8 a^{4} + 36 a^{3} + 38 a^{2} + 36 a + 49\right)\cdot 53^{6} + \left(38 a^{5} + 4 a^{4} + 46 a^{3} + 26 a^{2} + 42 a + 52\right)\cdot 53^{7} + \left(33 a^{5} + 6 a^{4} + 41 a^{3} + 2 a^{2} + 17 a + 27\right)\cdot 53^{8} + \left(27 a^{5} + 10 a^{4} + 10 a^{3} + 32 a^{2} + 31 a + 28\right)\cdot 53^{9} +O(53^{10})$ 27a^5 + 38a^4 + 24a^3 + 36a^2 + 4a + 15 + (33a^5 + 51a^4 + 44a^3 + 41a^2 + 28a + 26)53 + (49a^5 + 3a^4 + 4a^3 + 15a^2 + 52a + 48)53^2 + (13a^5 + 24a^4 + 42a^3 + 3a^2 + 36a + 9)53^3 + (4a^5 + 6a^4 + 16a^3 + 44a^2 + 24a + 29)53^4 + (18a^5 + 4a^4 + 47a^3 + 38a^2 + 3a + 27)53^5 + (2a^5 + 8a^4 + 36a^3 + 38a^2 + 36a + 49)53^6 + (38a^5 + 4a^4 + 46a^3 + 26a^2 + 42a + 52)53^7 + (33a^5 + 6a^4 + 41a^3 + 2a^2 + 17a + 27)53^8 + (27a^5 + 10a^4 + 10a^3 + 32a^2 + 31a + 28)*53^9+O(53^10)
$r_{ 5 }$	$=$	$ 11 a^{5} + 31 a^{4} + 17 a^{3} + 46 a^{2} + 41 a + 45 + \left(14 a^{5} + 40 a^{4} + 14 a^{3} + 46 a^{2} + 10 a\right)\cdot 53 + \left(37 a^{5} + 43 a^{4} + 16 a^{3} + 8 a + 41\right)\cdot 53^{2} + \left(3 a^{5} + 40 a^{4} + 35 a^{3} + 51 a^{2} + 31 a + 52\right)\cdot 53^{3} + \left(17 a^{5} + 29 a^{4} + 31 a^{3} + 40 a^{2} + 44 a + 18\right)\cdot 53^{4} + \left(10 a^{5} + 38 a^{4} + 12 a^{3} + 15 a^{2} + 3 a + 33\right)\cdot 53^{5} + \left(33 a^{5} + 52 a^{3} + 10 a^{2} + 6 a + 46\right)\cdot 53^{6} + \left(3 a^{5} + 6 a^{4} + 36 a^{3} + 46 a^{2} + 23 a + 52\right)\cdot 53^{7} + \left(47 a^{5} + 42 a^{4} + 2 a^{3} + 47 a^{2} + 44 a + 32\right)\cdot 53^{8} + \left(33 a^{5} + 23 a^{4} + 21 a^{3} + 42 a^{2} + a + 50\right)\cdot 53^{9} +O(53^{10})$ 11a^5 + 31a^4 + 17a^3 + 46a^2 + 41a + 45 + (14a^5 + 40a^4 + 14a^3 + 46a^2 + 10a)53 + (37a^5 + 43a^4 + 16a^3 + 8a + 41)53^2 + (3a^5 + 40a^4 + 35a^3 + 51a^2 + 31a + 52)53^3 + (17a^5 + 29a^4 + 31a^3 + 40a^2 + 44a + 18)53^4 + (10a^5 + 38a^4 + 12a^3 + 15a^2 + 3a + 33)53^5 + (33a^5 + 52a^3 + 10a^2 + 6a + 46)53^6 + (3a^5 + 6a^4 + 36a^3 + 46a^2 + 23a + 52)53^7 + (47a^5 + 42a^4 + 2a^3 + 47a^2 + 44a + 32)53^8 + (33a^5 + 23a^4 + 21a^3 + 42a^2 + a + 50)53^9+O(53^10)
$r_{ 6 }$	$=$	$ 43 a^{5} + 47 a^{4} + 10 a^{3} + 47 a^{2} + 6 a + 38 + \left(37 a^{5} + 10 a^{4} + a^{3} + 7 a^{2} + 11 a + 1\right)\cdot 53 + \left(51 a^{5} + 46 a^{4} + 21 a^{3} + 35 a^{2} + 17 a + 11\right)\cdot 53^{2} + \left(14 a^{5} + 6 a^{4} + 35 a^{3} + 20 a^{2} + 21 a + 29\right)\cdot 53^{3} + \left(41 a^{5} + 30 a^{4} + 22 a^{3} + 42 a^{2} + 32 a + 13\right)\cdot 53^{4} + \left(39 a^{5} + 18 a^{4} + 17 a^{3} + 12 a^{2} + 7 a + 43\right)\cdot 53^{5} + \left(26 a^{5} + 45 a^{4} + 12 a^{3} + 18 a^{2} + 52 a + 12\right)\cdot 53^{6} + \left(26 a^{5} + 49 a^{4} + 50 a^{3} + 51 a^{2} + 13 a\right)\cdot 53^{7} + \left(15 a^{5} + 9 a^{4} + 2 a^{3} + 37 a^{2} + 31 a + 7\right)\cdot 53^{8} + \left(33 a^{4} + 34 a^{3} + 36 a + 47\right)\cdot 53^{9} +O(53^{10})$ 43a^5 + 47a^4 + 10a^3 + 47a^2 + 6a + 38 + (37a^5 + 10a^4 + a^3 + 7a^2 + 11a + 1)53 + (51a^5 + 46a^4 + 21a^3 + 35a^2 + 17a + 11)53^2 + (14a^5 + 6a^4 + 35a^3 + 20a^2 + 21a + 29)53^3 + (41a^5 + 30a^4 + 22a^3 + 42a^2 + 32a + 13)53^4 + (39a^5 + 18a^4 + 17a^3 + 12a^2 + 7a + 43)53^5 + (26a^5 + 45a^4 + 12a^3 + 18a^2 + 52a + 12)53^6 + (26a^5 + 49a^4 + 50a^3 + 51a^2 + 13a)53^7 + (15a^5 + 9a^4 + 2a^3 + 37a^2 + 31a + 7)53^8 + (33a^4 + 34a^3 + 36a + 47)53^9+O(53^10)
$r_{ 7 }$	$=$	$ 11 a^{5} + 49 a^{4} + 34 a^{3} + 51 a^{2} + 19 a + 51 + \left(18 a^{5} + 22 a^{4} + 11 a^{3} + 33 a^{2} + 17 a + 1\right)\cdot 53 + \left(35 a^{5} + 19 a^{4} + 42 a^{3} + 9 a^{2} + 14 a + 17\right)\cdot 53^{2} + \left(42 a^{5} + 12 a^{4} + 32 a^{3} + 3 a^{2} + 39 a + 38\right)\cdot 53^{3} + \left(27 a^{5} + 22 a^{4} + 18 a^{3} + 42 a^{2} + 38 a + 48\right)\cdot 53^{4} + \left(35 a^{5} + 27 a^{4} + 6 a^{2} + 36 a + 52\right)\cdot 53^{5} + \left(18 a^{5} + 17 a^{4} + 8 a^{3} + 30 a^{2} + 17 a + 42\right)\cdot 53^{6} + \left(33 a^{5} + 18 a^{4} + 11 a^{3} + 24 a^{2} + 43 a + 50\right)\cdot 53^{7} + \left(15 a^{5} + 11 a^{4} + 2 a^{3} + 12 a^{2} + 31 a + 20\right)\cdot 53^{8} + \left(46 a^{5} + 17 a^{4} + 28 a^{3} + 2 a^{2} + 2 a + 50\right)\cdot 53^{9} +O(53^{10})$ 11a^5 + 49a^4 + 34a^3 + 51a^2 + 19a + 51 + (18a^5 + 22a^4 + 11a^3 + 33a^2 + 17a + 1)53 + (35a^5 + 19a^4 + 42a^3 + 9a^2 + 14a + 17)53^2 + (42a^5 + 12a^4 + 32a^3 + 3a^2 + 39a + 38)53^3 + (27a^5 + 22a^4 + 18a^3 + 42a^2 + 38a + 48)53^4 + (35a^5 + 27a^4 + 6a^2 + 36a + 52)53^5 + (18a^5 + 17a^4 + 8a^3 + 30a^2 + 17a + 42)53^6 + (33a^5 + 18a^4 + 11a^3 + 24a^2 + 43a + 50)53^7 + (15a^5 + 11a^4 + 2a^3 + 12a^2 + 31a + 20)53^8 + (46a^5 + 17a^4 + 28a^3 + 2a^2 + 2a + 50)53^9+O(53^10)
$r_{ 8 }$	$=$	$ 44 a^{5} + 37 a^{4} + 30 a^{3} + 45 a^{2} + 23 a + 45 + \left(22 a^{5} + 8 a^{4} + 36 a^{3} + 14 a^{2} + 26 a + 45\right)\cdot 53 + \left(43 a^{5} + 24 a^{4} + 39 a^{3} + 8 a^{2} + 6 a + 9\right)\cdot 53^{2} + \left(2 a^{5} + 2 a^{4} + 43 a^{3} + 28 a^{2} + 31 a + 10\right)\cdot 53^{3} + \left(10 a^{5} + 43 a^{4} + 37 a^{3} + 44 a^{2} + 37 a + 27\right)\cdot 53^{4} + \left(30 a^{5} + 26 a^{4} + 38 a^{3} + 51 a^{2} + 18 a + 49\right)\cdot 53^{5} + \left(8 a^{5} + 35 a^{4} + 16 a^{3} + 24 a^{2} + 47 a + 5\right)\cdot 53^{6} + \left(13 a^{5} + 49 a^{4} + 9 a^{3} + 42 a^{2} + 17 a + 14\right)\cdot 53^{7} + \left(26 a^{5} + 8 a^{4} + 44 a^{3} + 46 a^{2} + 40 a + 15\right)\cdot 53^{8} + \left(47 a^{5} + 10 a^{4} + 10 a^{3} + a^{2} + 6 a + 29\right)\cdot 53^{9} +O(53^{10})$ 44a^5 + 37a^4 + 30a^3 + 45a^2 + 23a + 45 + (22a^5 + 8a^4 + 36a^3 + 14a^2 + 26a + 45)53 + (43a^5 + 24a^4 + 39a^3 + 8a^2 + 6a + 9)53^2 + (2a^5 + 2a^4 + 43a^3 + 28a^2 + 31a + 10)53^3 + (10a^5 + 43a^4 + 37a^3 + 44a^2 + 37a + 27)53^4 + (30a^5 + 26a^4 + 38a^3 + 51a^2 + 18a + 49)53^5 + (8a^5 + 35a^4 + 16a^3 + 24a^2 + 47a + 5)53^6 + (13a^5 + 49a^4 + 9a^3 + 42a^2 + 17a + 14)53^7 + (26a^5 + 8a^4 + 44a^3 + 46a^2 + 40a + 15)53^8 + (47a^5 + 10a^4 + 10a^3 + a^2 + 6a + 29)53^9+O(53^10)
$r_{ 9 }$	$=$	$ 38 a^{5} + 52 a^{4} + 45 a^{3} + 20 a^{2} + 44 a + 49 + \left(9 a^{5} + 34 a^{4} + 39 a^{3} + 15 a^{2} + 43 a + 3\right)\cdot 53 + \left(a^{5} + 52 a^{4} + 20 a^{3} + 20 a^{2} + 42 a + 10\right)\cdot 53^{2} + \left(14 a^{5} + 30 a^{4} + 33 a^{3} + 29 a^{2} + 30 a + 51\right)\cdot 53^{3} + \left(6 a^{5} + 39 a^{4} + 14 a^{3} + 45 a^{2} + 49 a + 47\right)\cdot 53^{4} + \left(16 a^{5} + 44 a^{4} + 19 a^{3} + 48 a^{2} + 42 a + 44\right)\cdot 53^{5} + \left(19 a^{5} + 6 a^{4} + 25 a^{3} + 29 a^{2} + 48 a + 45\right)\cdot 53^{6} + \left(39 a^{5} + 43 a^{4} + 47 a^{3} + 33 a^{2} + 28 a + 22\right)\cdot 53^{7} + \left(35 a^{5} + 37 a^{4} + 42 a^{3} + 3 a^{2} + 45 a + 46\right)\cdot 53^{8} + \left(50 a^{5} + 41 a^{4} + 32 a^{3} + 52 a^{2} + 30 a + 49\right)\cdot 53^{9} +O(53^{10})$ 38a^5 + 52a^4 + 45a^3 + 20a^2 + 44a + 49 + (9a^5 + 34a^4 + 39a^3 + 15a^2 + 43a + 3)53 + (a^5 + 52a^4 + 20a^3 + 20a^2 + 42a + 10)53^2 + (14a^5 + 30a^4 + 33a^3 + 29a^2 + 30a + 51)53^3 + (6a^5 + 39a^4 + 14a^3 + 45a^2 + 49a + 47)53^4 + (16a^5 + 44a^4 + 19a^3 + 48a^2 + 42a + 44)53^5 + (19a^5 + 6a^4 + 25a^3 + 29a^2 + 48a + 45)53^6 + (39a^5 + 43a^4 + 47a^3 + 33a^2 + 28a + 22)53^7 + (35a^5 + 37a^4 + 42a^3 + 3a^2 + 45a + 46)53^8 + (50a^5 + 41a^4 + 32a^3 + 52a^2 + 30a + 49)53^9+O(53^10)
$r_{ 10 }$	$=$	$ 17 a^{5} + 7 a^{4} + a^{3} + 7 a^{2} + 7 a + 4 + \left(43 a^{5} + 2 a^{4} + 44 a^{3} + 29 a^{2} + 31 a + 30\right)\cdot 53 + \left(25 a^{5} + 24 a^{4} + 20 a^{3} + 10 a^{2} + 49 a + 22\right)\cdot 53^{2} + \left(48 a^{5} + 52 a^{4} + 7 a^{3} + 5 a^{2} + 13 a + 42\right)\cdot 53^{3} + \left(a^{5} + 2 a^{4} + 9 a^{3} + 48 a^{2} + 46 a\right)\cdot 53^{4} + \left(32 a^{5} + 27 a^{4} + 51 a^{2} + 30 a + 9\right)\cdot 53^{5} + \left(37 a^{5} + 27 a^{4} + 22 a^{3} + 3 a^{2} + 38 a + 17\right)\cdot 53^{6} + \left(21 a^{5} + 49 a^{4} + 8 a^{3} + 37 a^{2} + 35 a + 23\right)\cdot 53^{7} + \left(29 a^{5} + 4 a^{4} + 31 a^{3} + 4 a^{2} + 31 a + 46\right)\cdot 53^{8} + \left(5 a^{5} + 25 a^{4} + 38 a^{3} + 3 a^{2} + 47 a + 34\right)\cdot 53^{9} +O(53^{10})$ 17a^5 + 7a^4 + a^3 + 7a^2 + 7a + 4 + (43a^5 + 2a^4 + 44a^3 + 29a^2 + 31a + 30)53 + (25a^5 + 24a^4 + 20a^3 + 10a^2 + 49a + 22)53^2 + (48a^5 + 52a^4 + 7a^3 + 5a^2 + 13a + 42)53^3 + (a^5 + 2a^4 + 9a^3 + 48a^2 + 46a)53^4 + (32a^5 + 27a^4 + 51a^2 + 30a + 9)53^5 + (37a^5 + 27a^4 + 22a^3 + 3a^2 + 38a + 17)53^6 + (21a^5 + 49a^4 + 8a^3 + 37a^2 + 35a + 23)53^7 + (29a^5 + 4a^4 + 31a^3 + 4a^2 + 31a + 46)53^8 + (5a^5 + 25a^4 + 38a^3 + 3a^2 + 47a + 34)53^9+O(53^10)
$r_{ 11 }$	$=$	$ 30 a^{5} + 36 a^{4} + 35 a^{3} + 11 a^{2} + 29 a + 3 + \left(38 a^{5} + 14 a^{4} + 34 a^{3} + 3 a^{2} + 19 a + 44\right)\cdot 53 + \left(16 a^{5} + 23 a^{4} + 9 a^{3} + 45 a^{2} + 39 a + 49\right)\cdot 53^{2} + \left(2 a^{5} + a^{4} + 33 a^{3} + 40 a^{2} + 39 a + 20\right)\cdot 53^{3} + \left(18 a^{5} + 38 a^{4} + 27 a^{3} + 31 a^{2} + 13 a + 25\right)\cdot 53^{4} + \left(49 a^{5} + 17 a^{4} + 36 a^{3} + 28 a^{2} + 14 a + 27\right)\cdot 53^{5} + \left(31 a^{5} + 49 a^{4} + 42 a^{2} + 42 a + 47\right)\cdot 53^{6} + \left(2 a^{5} + 42 a^{4} + 16 a^{3} + 31 a^{2} + 44 a + 1\right)\cdot 53^{7} + \left(a^{5} + 28 a^{4} + 42 a^{3} + 23 a^{2} + 52 a + 9\right)\cdot 53^{8} + \left(7 a^{5} + 24 a^{4} + 48 a^{3} + 22 a^{2} + 2 a + 16\right)\cdot 53^{9} +O(53^{10})$ 30a^5 + 36a^4 + 35a^3 + 11a^2 + 29a + 3 + (38a^5 + 14a^4 + 34a^3 + 3a^2 + 19a + 44)53 + (16a^5 + 23a^4 + 9a^3 + 45a^2 + 39a + 49)53^2 + (2a^5 + a^4 + 33a^3 + 40a^2 + 39a + 20)53^3 + (18a^5 + 38a^4 + 27a^3 + 31a^2 + 13a + 25)53^4 + (49a^5 + 17a^4 + 36a^3 + 28a^2 + 14a + 27)53^5 + (31a^5 + 49a^4 + 42a^2 + 42a + 47)53^6 + (2a^5 + 42a^4 + 16a^3 + 31a^2 + 44a + 1)53^7 + (a^5 + 28a^4 + 42a^3 + 23a^2 + 52a + 9)53^8 + (7a^5 + 24a^4 + 48a^3 + 22a^2 + 2a + 16)53^9+O(53^10)
$r_{ 12 }$	$=$	$ 26 a^{5} + 15 a^{4} + 29 a^{3} + 17 a^{2} + 49 a + 38 + \left(19 a^{5} + a^{4} + 8 a^{3} + 11 a^{2} + 24 a + 26\right)\cdot 53 + \left(3 a^{5} + 49 a^{4} + 48 a^{3} + 37 a^{2} + 4\right)\cdot 53^{2} + \left(39 a^{5} + 28 a^{4} + 10 a^{3} + 49 a^{2} + 16 a + 43\right)\cdot 53^{3} + \left(48 a^{5} + 46 a^{4} + 36 a^{3} + 8 a^{2} + 28 a + 23\right)\cdot 53^{4} + \left(34 a^{5} + 48 a^{4} + 5 a^{3} + 14 a^{2} + 49 a + 25\right)\cdot 53^{5} + \left(50 a^{5} + 44 a^{4} + 16 a^{3} + 14 a^{2} + 16 a + 3\right)\cdot 53^{6} + \left(14 a^{5} + 48 a^{4} + 6 a^{3} + 26 a^{2} + 10 a\right)\cdot 53^{7} + \left(19 a^{5} + 46 a^{4} + 11 a^{3} + 50 a^{2} + 35 a + 25\right)\cdot 53^{8} + \left(25 a^{5} + 42 a^{4} + 42 a^{3} + 20 a^{2} + 21 a + 24\right)\cdot 53^{9} +O(53^{10})$ 26a^5 + 15a^4 + 29a^3 + 17a^2 + 49a + 38 + (19a^5 + a^4 + 8a^3 + 11a^2 + 24a + 26)53 + (3a^5 + 49a^4 + 48a^3 + 37a^2 + 4)53^2 + (39a^5 + 28a^4 + 10a^3 + 49a^2 + 16a + 43)53^3 + (48a^5 + 46a^4 + 36a^3 + 8a^2 + 28a + 23)53^4 + (34a^5 + 48a^4 + 5a^3 + 14a^2 + 49a + 25)53^5 + (50a^5 + 44a^4 + 16a^3 + 14a^2 + 16a + 3)53^6 + (14a^5 + 48a^4 + 6a^3 + 26a^2 + 10a)53^7 + (19a^5 + 46a^4 + 11a^3 + 50a^2 + 35a + 25)53^8 + (25a^5 + 42a^4 + 42a^3 + 20a^2 + 21a + 24)*53^9+O(53^10)
$r_{ 13 }$	$=$	$ 42 a^{5} + 22 a^{4} + 36 a^{3} + 7 a^{2} + 12 a + 8 + \left(38 a^{5} + 12 a^{4} + 38 a^{3} + 6 a^{2} + 42 a + 52\right)\cdot 53 + \left(15 a^{5} + 9 a^{4} + 36 a^{3} + 52 a^{2} + 44 a + 11\right)\cdot 53^{2} + \left(49 a^{5} + 12 a^{4} + 17 a^{3} + a^{2} + 21 a\right)\cdot 53^{3} + \left(35 a^{5} + 23 a^{4} + 21 a^{3} + 12 a^{2} + 8 a + 34\right)\cdot 53^{4} + \left(42 a^{5} + 14 a^{4} + 40 a^{3} + 37 a^{2} + 49 a + 19\right)\cdot 53^{5} + \left(19 a^{5} + 52 a^{4} + 42 a^{2} + 46 a + 6\right)\cdot 53^{6} + \left(49 a^{5} + 46 a^{4} + 16 a^{3} + 6 a^{2} + 29 a\right)\cdot 53^{7} + \left(5 a^{5} + 10 a^{4} + 50 a^{3} + 5 a^{2} + 8 a + 20\right)\cdot 53^{8} + \left(19 a^{5} + 29 a^{4} + 31 a^{3} + 10 a^{2} + 51 a + 2\right)\cdot 53^{9} +O(53^{10})$ 42a^5 + 22a^4 + 36a^3 + 7a^2 + 12a + 8 + (38a^5 + 12a^4 + 38a^3 + 6a^2 + 42a + 52)53 + (15a^5 + 9a^4 + 36a^3 + 52a^2 + 44a + 11)53^2 + (49a^5 + 12a^4 + 17a^3 + a^2 + 21a)53^3 + (35a^5 + 23a^4 + 21a^3 + 12a^2 + 8a + 34)53^4 + (42a^5 + 14a^4 + 40a^3 + 37a^2 + 49a + 19)53^5 + (19a^5 + 52a^4 + 42a^2 + 46a + 6)53^6 + (49a^5 + 46a^4 + 16a^3 + 6a^2 + 29a)53^7 + (5a^5 + 10a^4 + 50a^3 + 5a^2 + 8a + 20)53^8 + (19a^5 + 29a^4 + 31a^3 + 10a^2 + 51a + 2)*53^9+O(53^10)
$r_{ 14 }$	$=$	$ 10 a^{5} + 6 a^{4} + 43 a^{3} + 6 a^{2} + 47 a + 15 + \left(15 a^{5} + 42 a^{4} + 51 a^{3} + 45 a^{2} + 41 a + 51\right)\cdot 53 + \left(a^{5} + 6 a^{4} + 31 a^{3} + 17 a^{2} + 35 a + 41\right)\cdot 53^{2} + \left(38 a^{5} + 46 a^{4} + 17 a^{3} + 32 a^{2} + 31 a + 23\right)\cdot 53^{3} + \left(11 a^{5} + 22 a^{4} + 30 a^{3} + 10 a^{2} + 20 a + 39\right)\cdot 53^{4} + \left(13 a^{5} + 34 a^{4} + 35 a^{3} + 40 a^{2} + 45 a + 9\right)\cdot 53^{5} + \left(26 a^{5} + 7 a^{4} + 40 a^{3} + 34 a^{2} + 40\right)\cdot 53^{6} + \left(26 a^{5} + 3 a^{4} + 2 a^{3} + a^{2} + 39 a + 52\right)\cdot 53^{7} + \left(37 a^{5} + 43 a^{4} + 50 a^{3} + 15 a^{2} + 21 a + 45\right)\cdot 53^{8} + \left(52 a^{5} + 19 a^{4} + 18 a^{3} + 52 a^{2} + 16 a + 5\right)\cdot 53^{9} +O(53^{10})$ 10a^5 + 6a^4 + 43a^3 + 6a^2 + 47a + 15 + (15a^5 + 42a^4 + 51a^3 + 45a^2 + 41a + 51)53 + (a^5 + 6a^4 + 31a^3 + 17a^2 + 35a + 41)53^2 + (38a^5 + 46a^4 + 17a^3 + 32a^2 + 31a + 23)53^3 + (11a^5 + 22a^4 + 30a^3 + 10a^2 + 20a + 39)53^4 + (13a^5 + 34a^4 + 35a^3 + 40a^2 + 45a + 9)53^5 + (26a^5 + 7a^4 + 40a^3 + 34a^2 + 40)53^6 + (26a^5 + 3a^4 + 2a^3 + a^2 + 39a + 52)53^7 + (37a^5 + 43a^4 + 50a^3 + 15a^2 + 21a + 45)53^8 + (52a^5 + 19a^4 + 18a^3 + 52a^2 + 16a + 5)53^9+O(53^10)
$r_{ 15 }$	$=$	$ 42 a^{5} + 4 a^{4} + 19 a^{3} + 2 a^{2} + 34 a + 2 + \left(34 a^{5} + 30 a^{4} + 41 a^{3} + 19 a^{2} + 35 a + 51\right)\cdot 53 + \left(17 a^{5} + 33 a^{4} + 10 a^{3} + 43 a^{2} + 38 a + 35\right)\cdot 53^{2} + \left(10 a^{5} + 40 a^{4} + 20 a^{3} + 49 a^{2} + 13 a + 14\right)\cdot 53^{3} + \left(25 a^{5} + 30 a^{4} + 34 a^{3} + 10 a^{2} + 14 a + 4\right)\cdot 53^{4} + \left(17 a^{5} + 25 a^{4} + 52 a^{3} + 46 a^{2} + 16 a\right)\cdot 53^{5} + \left(34 a^{5} + 35 a^{4} + 44 a^{3} + 22 a^{2} + 35 a + 10\right)\cdot 53^{6} + \left(19 a^{5} + 34 a^{4} + 41 a^{3} + 28 a^{2} + 9 a + 2\right)\cdot 53^{7} + \left(37 a^{5} + 41 a^{4} + 50 a^{3} + 40 a^{2} + 21 a + 32\right)\cdot 53^{8} + \left(6 a^{5} + 35 a^{4} + 24 a^{3} + 50 a^{2} + 50 a + 2\right)\cdot 53^{9} +O(53^{10})$ 42a^5 + 4a^4 + 19a^3 + 2a^2 + 34a + 2 + (34a^5 + 30a^4 + 41a^3 + 19a^2 + 35a + 51)53 + (17a^5 + 33a^4 + 10a^3 + 43a^2 + 38a + 35)53^2 + (10a^5 + 40a^4 + 20a^3 + 49a^2 + 13a + 14)53^3 + (25a^5 + 30a^4 + 34a^3 + 10a^2 + 14a + 4)53^4 + (17a^5 + 25a^4 + 52a^3 + 46a^2 + 16a)53^5 + (34a^5 + 35a^4 + 44a^3 + 22a^2 + 35a + 10)53^6 + (19a^5 + 34a^4 + 41a^3 + 28a^2 + 9a + 2)53^7 + (37a^5 + 41a^4 + 50a^3 + 40a^2 + 21a + 32)53^8 + (6a^5 + 35a^4 + 24a^3 + 50a^2 + 50a + 2)*53^9+O(53^10)
$r_{ 16 }$	$=$	$ 9 a^{5} + 16 a^{4} + 23 a^{3} + 8 a^{2} + 30 a + 8 + \left(30 a^{5} + 44 a^{4} + 16 a^{3} + 38 a^{2} + 26 a + 7\right)\cdot 53 + \left(9 a^{5} + 28 a^{4} + 13 a^{3} + 44 a^{2} + 46 a + 43\right)\cdot 53^{2} + \left(50 a^{5} + 50 a^{4} + 9 a^{3} + 24 a^{2} + 21 a + 42\right)\cdot 53^{3} + \left(42 a^{5} + 9 a^{4} + 15 a^{3} + 8 a^{2} + 15 a + 25\right)\cdot 53^{4} + \left(22 a^{5} + 26 a^{4} + 14 a^{3} + a^{2} + 34 a + 3\right)\cdot 53^{5} + \left(44 a^{5} + 17 a^{4} + 36 a^{3} + 28 a^{2} + 5 a + 47\right)\cdot 53^{6} + \left(39 a^{5} + 3 a^{4} + 43 a^{3} + 10 a^{2} + 35 a + 38\right)\cdot 53^{7} + \left(26 a^{5} + 44 a^{4} + 8 a^{3} + 6 a^{2} + 12 a + 37\right)\cdot 53^{8} + \left(5 a^{5} + 42 a^{4} + 42 a^{3} + 51 a^{2} + 46 a + 23\right)\cdot 53^{9} +O(53^{10})$ 9a^5 + 16a^4 + 23a^3 + 8a^2 + 30a + 8 + (30a^5 + 44a^4 + 16a^3 + 38a^2 + 26a + 7)53 + (9a^5 + 28a^4 + 13a^3 + 44a^2 + 46a + 43)53^2 + (50a^5 + 50a^4 + 9a^3 + 24a^2 + 21a + 42)53^3 + (42a^5 + 9a^4 + 15a^3 + 8a^2 + 15a + 25)53^4 + (22a^5 + 26a^4 + 14a^3 + a^2 + 34a + 3)53^5 + (44a^5 + 17a^4 + 36a^3 + 28a^2 + 5a + 47)53^6 + (39a^5 + 3a^4 + 43a^3 + 10a^2 + 35a + 38)53^7 + (26a^5 + 44a^4 + 8a^3 + 6a^2 + 12a + 37)53^8 + (5a^5 + 42a^4 + 42a^3 + 51a^2 + 46a + 23)53^9+O(53^10)

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

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

Character values on conjugacy classes

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

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Hilbert	Bianchi

Elliptic curves over $\Q$
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Defining polynomial

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	2.1184.16t60.a.a
Dimension	$2$
Group	$\SL(2,3):C_2$
Conductor	$1184$
Root number	not computed
Indicator	$0$