LMFDB - Artin representation 2.24843.24t21.a.b

Basic invariants

Dimension:	$2$
Group:	16T60
Conductor:	$24843$$\medspace = 3 \cdot 7^{2} \cdot 13^{2} $
Artin stem field:	Galois closure of 16.0.30853268336830129281.2
Galois orbit size:	$2$
Smallest permutation container:	24T21
Parity:	odd
Determinant:	1.3.2t1.a.a
Projective image:	$A_4$
Projective stem field:	Galois closure of 4.0.74529.1

Defining polynomial

$f(x)$

$=$

$ x^{16} - 5 x^{15} + 31 x^{13} - 8 x^{12} - 101 x^{11} + 7 x^{10} + 183 x^{9} + 32 x^{8} - 183 x^{7} + \cdots + 7 $

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $ x^{6} + 35x^{3} + 4x^{2} + 30x + 2 $ x^6 + 35*x^3 + 4*x^2 + 30*x + 2

Roots:

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

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

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

Character values on conjugacy classes

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

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	2.24843.24t21.a.b
Dimension	$2$
Group	$\SL(2,3):C_2$
Conductor	$24843$
Root number	not computed
Indicator	$0$