LMFDB - Artin representation 2.3364.14t3.a.a

Basic invariants

Dimension:	$2$
Group:	$D_{14}$
Conductor:	$3364$$\medspace = 2^{2} \cdot 29^{2} $
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin stem field:	Galois closure of 14.2.42027535208278434566144.1
Galois orbit size:	$3$
Smallest permutation container:	$D_{14}$
Parity:	odd
Determinant:	1.4.2t1.a.a
Projective image:	$D_7$
Projective stem field:	Galois closure of 7.1.38068692544.1

Defining polynomial

$f(x)$

$=$

$ x^{14} - x^{13} + 16 x^{12} - 17 x^{11} + 37 x^{10} - 258 x^{9} + 183 x^{8} - 141 x^{7} + 674 x^{6} + \cdots + 144 $

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^{7} + 9x + 51 $ x^7 + 9*x + 51

Roots:

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

Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.

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Generators of the action on the roots $r_1, \ldots, r_{ 14 }$

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	2.3364.14t3.a.a
Dimension	$2$
Group	$D_{14}$
Conductor	$3364$
Root number	$1$
Indicator	$1$