LMFDB - Galois orbit of Artin representations 2.403.16t60.a

Basic invariants

Dimension:	$2$
Group:	16T60
Conductor:	$403$$\medspace = 13 \cdot 31 $
Artin number field:	Galois closure of 16.0.695729420906184924961.1
Galois orbit size:	$4$
Smallest permutation container:	16T60
Parity:	odd
Projective image:	$A_4$
Projective field:	Galois closure of 4.0.162409.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 71 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 71 }$: $ x^{6} + x^{4} + 10x^{3} + 13x^{2} + 29x + 7 $

Roots:

$r_{ 1 }$	$=$	$ 8 a^{5} + 60 a^{4} + 66 a^{3} + 64 a^{2} + 40 a + 36 + \left(27 a^{5} + 50 a^{4} + 15 a^{3} + 66 a^{2} + 25 a + 43\right)\cdot 71 + \left(19 a^{5} + 64 a^{4} + 55 a^{3} + 56 a^{2} + 13 a + 33\right)\cdot 71^{2} + \left(36 a^{5} + 7 a^{4} + 48 a^{3} + 14 a^{2} + 65 a + 11\right)\cdot 71^{3} + \left(52 a^{5} + 50 a^{4} + 60 a^{3} + 55 a^{2} + 53 a + 1\right)\cdot 71^{4} + \left(63 a^{5} + 29 a^{4} + 32 a^{3} + 25 a^{2} + 44 a + 30\right)\cdot 71^{5} + \left(21 a^{5} + 61 a^{4} + 67 a^{3} + 43 a^{2} + 52 a + 33\right)\cdot 71^{6} + \left(27 a^{5} + 5 a^{4} + 65 a^{3} + 22 a^{2} + 31 a + 68\right)\cdot 71^{7} + \left(41 a^{5} + 65 a^{4} + 3 a^{3} + 47 a^{2} + 34 a + 9\right)\cdot 71^{8} + \left(8 a^{5} + 21 a^{4} + 40 a^{3} + 24 a^{2} + 52 a + 42\right)\cdot 71^{9} +O(71^{10})$ 8a^5 + 60a^4 + 66a^3 + 64a^2 + 40a + 36 + (27a^5 + 50a^4 + 15a^3 + 66a^2 + 25a + 43)71 + (19a^5 + 64a^4 + 55a^3 + 56a^2 + 13a + 33)71^2 + (36a^5 + 7a^4 + 48a^3 + 14a^2 + 65a + 11)71^3 + (52a^5 + 50a^4 + 60a^3 + 55a^2 + 53a + 1)71^4 + (63a^5 + 29a^4 + 32a^3 + 25a^2 + 44a + 30)71^5 + (21a^5 + 61a^4 + 67a^3 + 43a^2 + 52a + 33)71^6 + (27a^5 + 5a^4 + 65a^3 + 22a^2 + 31a + 68)71^7 + (41a^5 + 65a^4 + 3a^3 + 47a^2 + 34a + 9)71^8 + (8a^5 + 21a^4 + 40a^3 + 24a^2 + 52a + 42)*71^9+O(71^10)
$r_{ 2 }$	$=$	$ 67 a^{5} + 37 a^{4} + 43 a^{3} + 22 a^{2} + 9 a + 52 + \left(11 a^{5} + 68 a^{4} + 25 a^{3} + 17 a^{2} + 64 a + 42\right)\cdot 71 + \left(35 a^{5} + 53 a^{4} + a^{2} + 46 a + 12\right)\cdot 71^{2} + \left(37 a^{4} + 62 a^{3} + a^{2} + a + 35\right)\cdot 71^{3} + \left(48 a^{5} + 70 a^{4} + 13 a^{3} + 31 a^{2} + 9 a + 42\right)\cdot 71^{4} + \left(39 a^{5} + 28 a^{4} + 62 a^{3} + 53 a^{2} + 52 a + 60\right)\cdot 71^{5} + \left(35 a^{5} + a^{4} + 63 a^{3} + 25 a^{2} + 60 a + 66\right)\cdot 71^{6} + \left(11 a^{5} + 10 a^{4} + 19 a^{3} + 68 a^{2} + 17 a + 22\right)\cdot 71^{7} + \left(7 a^{5} + 5 a^{4} + 3 a^{3} + 65 a^{2} + 66 a + 65\right)\cdot 71^{8} + \left(38 a^{5} + 9 a^{4} + 15 a^{3} + 13 a^{2} + 27 a + 47\right)\cdot 71^{9} +O(71^{10})$ 67a^5 + 37a^4 + 43a^3 + 22a^2 + 9a + 52 + (11a^5 + 68a^4 + 25a^3 + 17a^2 + 64a + 42)71 + (35a^5 + 53a^4 + a^2 + 46a + 12)71^2 + (37a^4 + 62a^3 + a^2 + a + 35)71^3 + (48a^5 + 70a^4 + 13a^3 + 31a^2 + 9a + 42)71^4 + (39a^5 + 28a^4 + 62a^3 + 53a^2 + 52a + 60)71^5 + (35a^5 + a^4 + 63a^3 + 25a^2 + 60a + 66)71^6 + (11a^5 + 10a^4 + 19a^3 + 68a^2 + 17a + 22)71^7 + (7a^5 + 5a^4 + 3a^3 + 65a^2 + 66a + 65)71^8 + (38a^5 + 9a^4 + 15a^3 + 13a^2 + 27a + 47)*71^9+O(71^10)
$r_{ 3 }$	$=$	$ 53 a^{5} + 70 a^{4} + 68 a^{2} + 3 a + 18 + \left(37 a^{5} + 28 a^{4} + 38 a^{3} + 16 a^{2} + 11 a + 64\right)\cdot 71 + \left(52 a^{5} + 52 a^{4} + 4 a^{3} + 61 a^{2} + 37 a + 61\right)\cdot 71^{2} + \left(44 a^{5} + 63 a^{4} + 57 a^{3} + 35 a^{2} + 19 a + 43\right)\cdot 71^{3} + \left(49 a^{5} + 41 a^{4} + 33 a^{3} + 33 a^{2} + 23 a + 17\right)\cdot 71^{4} + \left(34 a^{5} + 50 a^{4} + 62 a^{3} + 20 a^{2} + 49 a + 56\right)\cdot 71^{5} + \left(22 a^{5} + 4 a^{4} + 15 a^{3} + 44 a^{2} + 31 a + 10\right)\cdot 71^{6} + \left(55 a^{5} + 45 a^{4} + 31 a^{3} + 5 a^{2} + 67 a + 32\right)\cdot 71^{7} + \left(33 a^{5} + 6 a^{4} + 20 a^{3} + 57 a^{2} + 26 a + 44\right)\cdot 71^{8} + \left(30 a^{5} + a^{4} + 26 a^{3} + 26 a^{2} + 52 a + 40\right)\cdot 71^{9} +O(71^{10})$ 53a^5 + 70a^4 + 68a^2 + 3a + 18 + (37a^5 + 28a^4 + 38a^3 + 16a^2 + 11a + 64)71 + (52a^5 + 52a^4 + 4a^3 + 61a^2 + 37a + 61)71^2 + (44a^5 + 63a^4 + 57a^3 + 35a^2 + 19a + 43)71^3 + (49a^5 + 41a^4 + 33a^3 + 33a^2 + 23a + 17)71^4 + (34a^5 + 50a^4 + 62a^3 + 20a^2 + 49a + 56)71^5 + (22a^5 + 4a^4 + 15a^3 + 44a^2 + 31a + 10)71^6 + (55a^5 + 45a^4 + 31a^3 + 5a^2 + 67a + 32)71^7 + (33a^5 + 6a^4 + 20a^3 + 57a^2 + 26a + 44)71^8 + (30a^5 + a^4 + 26a^3 + 26a^2 + 52a + 40)*71^9+O(71^10)
$r_{ 4 }$	$=$	$ 43 a^{5} + 35 a^{4} + 3 a^{3} + 13 a^{2} + 16 a + 41 + \left(30 a^{5} + 35 a^{4} + 13 a^{3} + 59 a^{2} + 64 a + 13\right)\cdot 71 + \left(9 a^{5} + 37 a^{4} + 34 a^{3} + 69 a^{2} + 9 a + 24\right)\cdot 71^{2} + \left(51 a^{4} + 15 a^{3} + 36 a^{2} + 37 a + 14\right)\cdot 71^{3} + \left(8 a^{5} + 11 a^{4} + 33 a^{3} + 25 a^{2} + 40 a + 16\right)\cdot 71^{4} + \left(40 a^{4} + 57 a^{3} + 47 a^{2} + 27 a + 51\right)\cdot 71^{5} + \left(53 a^{5} + 11 a^{4} + 14 a^{3} + 44 a^{2} + 32 a + 47\right)\cdot 71^{6} + \left(48 a^{5} + 26 a^{4} + 67 a^{3} + 16 a^{2} + 31 a + 10\right)\cdot 71^{7} + \left(66 a^{5} + 48 a^{4} + 51 a^{3} + 43 a^{2} + 41 a + 45\right)\cdot 71^{8} + \left(44 a^{5} + 31 a^{4} + 51 a^{3} + 58 a^{2} + 52 a + 34\right)\cdot 71^{9} +O(71^{10})$ 43a^5 + 35a^4 + 3a^3 + 13a^2 + 16a + 41 + (30a^5 + 35a^4 + 13a^3 + 59a^2 + 64a + 13)71 + (9a^5 + 37a^4 + 34a^3 + 69a^2 + 9a + 24)71^2 + (51a^4 + 15a^3 + 36a^2 + 37a + 14)71^3 + (8a^5 + 11a^4 + 33a^3 + 25a^2 + 40a + 16)71^4 + (40a^4 + 57a^3 + 47a^2 + 27a + 51)71^5 + (53a^5 + 11a^4 + 14a^3 + 44a^2 + 32a + 47)71^6 + (48a^5 + 26a^4 + 67a^3 + 16a^2 + 31a + 10)71^7 + (66a^5 + 48a^4 + 51a^3 + 43a^2 + 41a + 45)71^8 + (44a^5 + 31a^4 + 51a^3 + 58a^2 + 52a + 34)*71^9+O(71^10)
$r_{ 5 }$	$=$	$ 68 a^{5} + 66 a^{4} + 21 a^{3} + 14 a^{2} + 37 + \left(11 a^{5} + 43 a^{4} + 18 a^{3} + 60 a^{2} + 29 a + 39\right)\cdot 71 + \left(47 a^{5} + 38 a^{4} + 27 a^{3} + 54 a^{2} + 48 a + 44\right)\cdot 71^{2} + \left(34 a^{5} + 39 a^{4} + 15 a^{3} + 61 a^{2} + 48 a + 3\right)\cdot 71^{3} + \left(49 a^{5} + 62 a^{4} + 7 a^{3} + 61 a^{2} + 12 a + 64\right)\cdot 71^{4} + \left(68 a^{5} + 49 a^{4} + 65 a^{3} + 62 a^{2} + 41 a + 47\right)\cdot 71^{5} + \left(31 a^{5} + 37 a^{4} + 27 a^{3} + 9 a^{2} + 8 a + 46\right)\cdot 71^{6} + \left(57 a^{5} + 44 a^{4} + 36 a^{3} + 44 a^{2} + 11 a + 64\right)\cdot 71^{7} + \left(35 a^{5} + 31 a^{4} + 8 a^{3} + 49 a^{2} + 26 a + 68\right)\cdot 71^{8} + \left(43 a^{5} + 17 a^{4} + 9 a^{3} + 42 a^{2} + 42 a + 65\right)\cdot 71^{9} +O(71^{10})$ 68a^5 + 66a^4 + 21a^3 + 14a^2 + 37 + (11a^5 + 43a^4 + 18a^3 + 60a^2 + 29a + 39)71 + (47a^5 + 38a^4 + 27a^3 + 54a^2 + 48a + 44)71^2 + (34a^5 + 39a^4 + 15a^3 + 61a^2 + 48a + 3)71^3 + (49a^5 + 62a^4 + 7a^3 + 61a^2 + 12a + 64)71^4 + (68a^5 + 49a^4 + 65a^3 + 62a^2 + 41a + 47)71^5 + (31a^5 + 37a^4 + 27a^3 + 9a^2 + 8a + 46)71^6 + (57a^5 + 44a^4 + 36a^3 + 44a^2 + 11a + 64)71^7 + (35a^5 + 31a^4 + 8a^3 + 49a^2 + 26a + 68)71^8 + (43a^5 + 17a^4 + 9a^3 + 42a^2 + 42a + 65)71^9+O(71^10)
$r_{ 6 }$	$=$	$ 15 a^{5} + 55 a^{4} + 7 a^{3} + 35 a^{2} + 40 a + 52 + \left(44 a^{5} + 8 a^{4} + 30 a^{3} + 36 a^{2} + 44 a + 36\right)\cdot 71 + \left(46 a^{5} + 57 a^{4} + 15 a^{3} + 48 a^{2} + 32 a + 48\right)\cdot 71^{2} + \left(39 a^{5} + 15 a^{4} + 43 a^{3} + 50 a^{2} + 50 a + 59\right)\cdot 71^{3} + \left(49 a^{5} + 25 a^{4} + 30 a^{3} + 6 a^{2} + 52 a + 21\right)\cdot 71^{4} + \left(33 a^{5} + 10 a^{4} + 63 a^{3} + 44 a^{2} + 34 a + 8\right)\cdot 71^{5} + \left(24 a^{5} + 40 a^{4} + 15 a^{3} + 56 a^{2} + 19 a + 21\right)\cdot 71^{6} + \left(35 a^{5} + 30 a^{4} + 33 a^{3} + 38 a^{2} + 35 a + 51\right)\cdot 71^{7} + \left(26 a^{5} + 67 a^{4} + 52 a^{3} + 36 a^{2} + 66 a + 17\right)\cdot 71^{8} + \left(62 a^{5} + 9 a^{3} + 32 a^{2} + 42 a + 25\right)\cdot 71^{9} +O(71^{10})$ 15a^5 + 55a^4 + 7a^3 + 35a^2 + 40a + 52 + (44a^5 + 8a^4 + 30a^3 + 36a^2 + 44a + 36)71 + (46a^5 + 57a^4 + 15a^3 + 48a^2 + 32a + 48)71^2 + (39a^5 + 15a^4 + 43a^3 + 50a^2 + 50a + 59)71^3 + (49a^5 + 25a^4 + 30a^3 + 6a^2 + 52a + 21)71^4 + (33a^5 + 10a^4 + 63a^3 + 44a^2 + 34a + 8)71^5 + (24a^5 + 40a^4 + 15a^3 + 56a^2 + 19a + 21)71^6 + (35a^5 + 30a^4 + 33a^3 + 38a^2 + 35a + 51)71^7 + (26a^5 + 67a^4 + 52a^3 + 36a^2 + 66a + 17)71^8 + (62a^5 + 9a^3 + 32a^2 + 42a + 25)71^9+O(71^10)
$r_{ 7 }$	$=$	$ 7 a^{5} + 24 a^{4} + 49 a^{3} + 68 a^{2} + 2 a + 24 + \left(49 a^{5} + 31 a^{4} + 54 a^{3} + 63 a^{2} + 32 a + 66\right)\cdot 71 + \left(68 a^{5} + a^{4} + 26 a^{3} + 38 a^{2} + 51 a + 17\right)\cdot 71^{2} + \left(45 a^{5} + 47 a^{4} + 59 a^{3} + 46 a^{2} + 14 a + 6\right)\cdot 71^{3} + \left(23 a^{5} + 13 a^{4} + 47 a^{3} + 52 a^{2} + 12 a\right)\cdot 71^{4} + \left(55 a^{5} + 37 a^{4} + 5 a^{3} + 41 a^{2} + 29 a + 47\right)\cdot 71^{5} + \left(2 a^{5} + 67 a^{4} + 33 a^{3} + 13 a^{2} + 9 a + 38\right)\cdot 71^{6} + \left(26 a^{5} + 62 a^{4} + 27 a^{3} + 28 a^{2} + 53 a + 28\right)\cdot 71^{7} + \left(23 a^{5} + 36 a^{4} + 69 a^{3} + 15 a^{2} + 8 a + 46\right)\cdot 71^{8} + \left(60 a^{5} + 35 a^{4} + 65 a^{3} + 46 a^{2} + 11\right)\cdot 71^{9} +O(71^{10})$ 7a^5 + 24a^4 + 49a^3 + 68a^2 + 2a + 24 + (49a^5 + 31a^4 + 54a^3 + 63a^2 + 32a + 66)71 + (68a^5 + a^4 + 26a^3 + 38a^2 + 51a + 17)71^2 + (45a^5 + 47a^4 + 59a^3 + 46a^2 + 14a + 6)71^3 + (23a^5 + 13a^4 + 47a^3 + 52a^2 + 12a)71^4 + (55a^5 + 37a^4 + 5a^3 + 41a^2 + 29a + 47)71^5 + (2a^5 + 67a^4 + 33a^3 + 13a^2 + 9a + 38)71^6 + (26a^5 + 62a^4 + 27a^3 + 28a^2 + 53a + 28)71^7 + (23a^5 + 36a^4 + 69a^3 + 15a^2 + 8a + 46)71^8 + (60a^5 + 35a^4 + 65a^3 + 46a^2 + 11)*71^9+O(71^10)
$r_{ 8 }$	$=$	$ 49 a^{5} + 65 a^{4} + 3 a^{3} + 25 a^{2} + 16 a + 35 + \left(26 a^{5} + 2 a^{4} + a^{3} + 38 a^{2} + 58 a + 15\right)\cdot 71 + \left(69 a^{5} + 47 a^{4} + 11 a^{3} + 59 a^{2} + 49 a + 34\right)\cdot 71^{2} + \left(34 a^{5} + 31 a^{4} + 11 a^{3} + 50 a^{2} + 63 a + 28\right)\cdot 71^{3} + \left(57 a^{5} + 22 a^{4} + 29 a^{3} + 26 a^{2} + 13 a + 5\right)\cdot 71^{4} + \left(65 a^{5} + 4 a^{4} + 40 a^{3} + 10 a^{2} + 54 a + 56\right)\cdot 71^{5} + \left(11 a^{5} + 34 a^{4} + 42 a^{3} + 42 a^{2} + 24 a + 6\right)\cdot 71^{6} + \left(16 a^{5} + 42 a^{4} + 62 a^{3} + 29 a^{2} + 10 a + 38\right)\cdot 71^{7} + \left(64 a^{5} + 7 a^{4} + 54 a^{3} + 38 a^{2} + 3 a + 34\right)\cdot 71^{8} + \left(63 a^{5} + 65 a^{4} + 54 a^{3} + 70 a^{2} + 3 a + 5\right)\cdot 71^{9} +O(71^{10})$ 49a^5 + 65a^4 + 3a^3 + 25a^2 + 16a + 35 + (26a^5 + 2a^4 + a^3 + 38a^2 + 58a + 15)71 + (69a^5 + 47a^4 + 11a^3 + 59a^2 + 49a + 34)71^2 + (34a^5 + 31a^4 + 11a^3 + 50a^2 + 63a + 28)71^3 + (57a^5 + 22a^4 + 29a^3 + 26a^2 + 13a + 5)71^4 + (65a^5 + 4a^4 + 40a^3 + 10a^2 + 54a + 56)71^5 + (11a^5 + 34a^4 + 42a^3 + 42a^2 + 24a + 6)71^6 + (16a^5 + 42a^4 + 62a^3 + 29a^2 + 10a + 38)71^7 + (64a^5 + 7a^4 + 54a^3 + 38a^2 + 3a + 34)71^8 + (63a^5 + 65a^4 + 54a^3 + 70a^2 + 3a + 5)71^9+O(71^10)
$r_{ 9 }$	$=$	$ 62 a^{5} + 35 a^{4} + 53 a^{3} + 12 a^{2} + 29 a + 56 + \left(43 a^{5} + 13 a^{4} + 21 a^{3} + 66 a^{2} + 41 a + 2\right)\cdot 71 + \left(27 a^{5} + 21 a^{4} + 32 a^{3} + 45 a^{2} + 29 a + 38\right)\cdot 71^{2} + \left(42 a^{5} + 55 a^{4} + 32 a^{3} + 60 a^{2} + 6 a + 34\right)\cdot 71^{3} + \left(48 a^{5} + 70 a^{4} + 8 a^{3} + 32 a^{2} + 26 a + 8\right)\cdot 71^{4} + \left(19 a^{5} + 2 a^{4} + 11 a^{3} + 36 a^{2} + 8 a + 46\right)\cdot 71^{5} + \left(28 a^{5} + 42 a^{4} + 68 a^{3} + 39 a^{2} + 57 a + 10\right)\cdot 71^{6} + \left(47 a^{5} + 39 a^{4} + 69 a^{3} + 46 a^{2} + 40 a + 21\right)\cdot 71^{7} + \left(23 a^{5} + 41 a^{3} + 15 a^{2} + 19 a + 22\right)\cdot 71^{8} + \left(63 a^{5} + 43 a^{4} + 38 a^{3} + 52 a^{2} + 41 a + 45\right)\cdot 71^{9} +O(71^{10})$ 62a^5 + 35a^4 + 53a^3 + 12a^2 + 29a + 56 + (43a^5 + 13a^4 + 21a^3 + 66a^2 + 41a + 2)71 + (27a^5 + 21a^4 + 32a^3 + 45a^2 + 29a + 38)71^2 + (42a^5 + 55a^4 + 32a^3 + 60a^2 + 6a + 34)71^3 + (48a^5 + 70a^4 + 8a^3 + 32a^2 + 26a + 8)71^4 + (19a^5 + 2a^4 + 11a^3 + 36a^2 + 8a + 46)71^5 + (28a^5 + 42a^4 + 68a^3 + 39a^2 + 57a + 10)71^6 + (47a^5 + 39a^4 + 69a^3 + 46a^2 + 40a + 21)71^7 + (23a^5 + 41a^3 + 15a^2 + 19a + 22)71^8 + (63a^5 + 43a^4 + 38a^3 + 52a^2 + 41a + 45)71^9+O(71^10)
$r_{ 10 }$	$=$	$ 4 a^{5} + 34 a^{4} + 28 a^{3} + 49 a^{2} + 62 a + 64 + \left(59 a^{5} + 2 a^{4} + 45 a^{3} + 53 a^{2} + 6 a + 29\right)\cdot 71 + \left(35 a^{5} + 17 a^{4} + 70 a^{3} + 69 a^{2} + 24 a + 30\right)\cdot 71^{2} + \left(70 a^{5} + 33 a^{4} + 8 a^{3} + 69 a^{2} + 69 a + 47\right)\cdot 71^{3} + \left(22 a^{5} + 57 a^{3} + 39 a^{2} + 61 a + 3\right)\cdot 71^{4} + \left(31 a^{5} + 42 a^{4} + 8 a^{3} + 17 a^{2} + 18 a + 30\right)\cdot 71^{5} + \left(35 a^{5} + 69 a^{4} + 7 a^{3} + 45 a^{2} + 10 a + 36\right)\cdot 71^{6} + \left(59 a^{5} + 60 a^{4} + 51 a^{3} + 2 a^{2} + 53 a + 4\right)\cdot 71^{7} + \left(63 a^{5} + 65 a^{4} + 67 a^{3} + 5 a^{2} + 4 a + 31\right)\cdot 71^{8} + \left(32 a^{5} + 61 a^{4} + 55 a^{3} + 57 a^{2} + 43 a + 21\right)\cdot 71^{9} +O(71^{10})$ 4a^5 + 34a^4 + 28a^3 + 49a^2 + 62a + 64 + (59a^5 + 2a^4 + 45a^3 + 53a^2 + 6a + 29)71 + (35a^5 + 17a^4 + 70a^3 + 69a^2 + 24a + 30)71^2 + (70a^5 + 33a^4 + 8a^3 + 69a^2 + 69a + 47)71^3 + (22a^5 + 57a^3 + 39a^2 + 61a + 3)71^4 + (31a^5 + 42a^4 + 8a^3 + 17a^2 + 18a + 30)71^5 + (35a^5 + 69a^4 + 7a^3 + 45a^2 + 10a + 36)71^6 + (59a^5 + 60a^4 + 51a^3 + 2a^2 + 53a + 4)71^7 + (63a^5 + 65a^4 + 67a^3 + 5a^2 + 4a + 31)71^8 + (32a^5 + 61a^4 + 55a^3 + 57a^2 + 43a + 21)71^9+O(71^10)
$r_{ 11 }$	$=$	$ 19 a^{5} + 29 a^{4} + 22 a^{3} + 70 a^{2} + 55 a + 6 + \left(54 a^{5} + 35 a^{4} + 53 a^{3} + 35 a^{2} + 51 a + 48\right)\cdot 71 + \left(17 a^{5} + 61 a^{4} + 62 a^{3} + 44 a^{2} + 43 a + 14\right)\cdot 71^{2} + \left(38 a^{5} + 27 a^{4} + 66 a^{3} + 54 a^{2} + 63 a + 5\right)\cdot 71^{3} + \left(33 a^{5} + 36 a^{4} + 44 a^{3} + 10 a^{2} + 35 a + 49\right)\cdot 71^{4} + \left(64 a^{5} + 41 a^{4} + 68 a^{3} + 64 a^{2} + 63 a + 47\right)\cdot 71^{5} + \left(31 a^{5} + 47 a^{4} + 40 a^{3} + 68 a^{2} + 34 a + 23\right)\cdot 71^{6} + \left(3 a^{5} + 26 a^{4} + 12 a^{3} + 68 a^{2} + 30 a + 62\right)\cdot 71^{7} + \left(53 a^{5} + 52 a^{4} + 19 a^{3} + 39 a^{2} + 65 a + 67\right)\cdot 71^{8} + \left(34 a^{5} + 30 a^{4} + 4 a^{3} + 29 a^{2} + 15 a + 44\right)\cdot 71^{9} +O(71^{10})$ 19a^5 + 29a^4 + 22a^3 + 70a^2 + 55a + 6 + (54a^5 + 35a^4 + 53a^3 + 35a^2 + 51a + 48)71 + (17a^5 + 61a^4 + 62a^3 + 44a^2 + 43a + 14)71^2 + (38a^5 + 27a^4 + 66a^3 + 54a^2 + 63a + 5)71^3 + (33a^5 + 36a^4 + 44a^3 + 10a^2 + 35a + 49)71^4 + (64a^5 + 41a^4 + 68a^3 + 64a^2 + 63a + 47)71^5 + (31a^5 + 47a^4 + 40a^3 + 68a^2 + 34a + 23)71^6 + (3a^5 + 26a^4 + 12a^3 + 68a^2 + 30a + 62)71^7 + (53a^5 + 52a^4 + 19a^3 + 39a^2 + 65a + 67)71^8 + (34a^5 + 30a^4 + 4a^3 + 29a^2 + 15a + 44)*71^9+O(71^10)
$r_{ 12 }$	$=$	$ 28 a^{5} + 17 a^{4} + 67 a^{3} + 59 a^{2} + 40 a + 48 + \left(61 a^{5} + 22 a^{4} + 44 a^{3} + 55 a^{2} + 65 a + 4\right)\cdot 71 + \left(36 a^{5} + 20 a^{4} + 49 a^{3} + 67 a^{2} + 42 a + 25\right)\cdot 71^{2} + \left(19 a^{5} + 32 a^{4} + 47 a^{3} + 57 a^{2} + 46 a + 21\right)\cdot 71^{3} + \left(34 a^{5} + 45 a^{4} + 43 a^{3} + 35 a^{2} + 27 a + 35\right)\cdot 71^{4} + \left(40 a^{5} + 13 a^{4} + 46 a^{3} + 28 a^{2} + 67 a + 6\right)\cdot 71^{5} + \left(51 a^{5} + 2 a^{4} + 35 a^{3} + 9 a^{2} + a + 63\right)\cdot 71^{6} + \left(13 a^{5} + 20 a^{4} + 41 a^{3} + 56 a^{2} + 68 a + 58\right)\cdot 71^{7} + \left(14 a^{5} + 5 a^{4} + 33 a^{3} + 45 a^{2} + 33 a + 53\right)\cdot 71^{8} + \left(21 a^{5} + 65 a^{4} + 57 a^{3} + 3 a^{2} + 38 a + 65\right)\cdot 71^{9} +O(71^{10})$ 28a^5 + 17a^4 + 67a^3 + 59a^2 + 40a + 48 + (61a^5 + 22a^4 + 44a^3 + 55a^2 + 65a + 4)71 + (36a^5 + 20a^4 + 49a^3 + 67a^2 + 42a + 25)71^2 + (19a^5 + 32a^4 + 47a^3 + 57a^2 + 46a + 21)71^3 + (34a^5 + 45a^4 + 43a^3 + 35a^2 + 27a + 35)71^4 + (40a^5 + 13a^4 + 46a^3 + 28a^2 + 67a + 6)71^5 + (51a^5 + 2a^4 + 35a^3 + 9a^2 + a + 63)71^6 + (13a^5 + 20a^4 + 41a^3 + 56a^2 + 68a + 58)71^7 + (14a^5 + 5a^4 + 33a^3 + 45a^2 + 33a + 53)71^8 + (21a^5 + 65a^4 + 57a^3 + 3a^2 + 38a + 65)71^9+O(71^10)
$r_{ 13 }$	$=$	$ 3 a^{5} + 24 a^{4} + 51 a^{3} + 56 a^{2} + 15 a + 17 + \left(38 a^{5} + 40 a^{4} + 65 a^{3} + 37 a^{2} + 54 a + 13\right)\cdot 71 + \left(48 a^{5} + 45 a^{4} + 30 a^{3} + 20 a^{2} + 40 a + 48\right)\cdot 71^{2} + \left(16 a^{5} + 18 a^{4} + 63 a^{3} + 56 a^{2} + 9 a + 31\right)\cdot 71^{3} + \left(50 a^{5} + 22 a^{4} + 57 a^{3} + 18 a^{2} + 61 a + 26\right)\cdot 71^{4} + \left(32 a^{5} + 38 a^{4} + 43 a^{3} + 3 a^{2} + 5 a + 36\right)\cdot 71^{5} + \left(5 a^{5} + 19 a^{4} + 63 a^{3} + 7 a^{2} + 28 a + 55\right)\cdot 71^{6} + \left(22 a^{5} + 51 a^{4} + 67 a^{3} + 25 a^{2} + 31 a + 7\right)\cdot 71^{7} + \left(25 a^{5} + 56 a^{4} + 47 a^{3} + 3 a^{2} + 40 a + 45\right)\cdot 71^{8} + \left(32 a^{5} + 27 a^{4} + 23 a^{3} + 37 a^{2} + 8 a + 46\right)\cdot 71^{9} +O(71^{10})$ 3a^5 + 24a^4 + 51a^3 + 56a^2 + 15a + 17 + (38a^5 + 40a^4 + 65a^3 + 37a^2 + 54a + 13)71 + (48a^5 + 45a^4 + 30a^3 + 20a^2 + 40a + 48)71^2 + (16a^5 + 18a^4 + 63a^3 + 56a^2 + 9a + 31)71^3 + (50a^5 + 22a^4 + 57a^3 + 18a^2 + 61a + 26)71^4 + (32a^5 + 38a^4 + 43a^3 + 3a^2 + 5a + 36)71^5 + (5a^5 + 19a^4 + 63a^3 + 7a^2 + 28a + 55)71^6 + (22a^5 + 51a^4 + 67a^3 + 25a^2 + 31a + 7)71^7 + (25a^5 + 56a^4 + 47a^3 + 3a^2 + 40a + 45)71^8 + (32a^5 + 27a^4 + 23a^3 + 37a^2 + 8a + 46)*71^9+O(71^10)
$r_{ 14 }$	$=$	$ 57 a^{5} + 63 a^{4} + 16 a^{3} + 31 a^{2} + 33 a + 70 + \left(26 a^{5} + 68 a^{4} + 3 a^{3} + 43 a^{2} + 30 a + 58\right)\cdot 71 + \left(48 a^{5} + 69 a^{4} + 39 a^{3} + 61 a^{2} + 66 a + 21\right)\cdot 71^{2} + \left(23 a^{5} + 62 a^{4} + 17 a^{3} + 15 a^{2} + 19 a + 36\right)\cdot 71^{3} + \left(62 a^{5} + 66 a^{4} + 42 a^{3} + 47 a^{2} + 9 a + 39\right)\cdot 71^{4} + \left(24 a^{5} + 27 a^{4} + 34 a^{3} + 35 a^{2} + 54 a + 57\right)\cdot 71^{5} + \left(67 a^{5} + 69 a^{4} + 61 a^{3} + 2 a^{2} + 12 a + 5\right)\cdot 71^{6} + \left(31 a^{5} + 65 a^{4} + 43 a^{3} + 34 a^{2} + 34 a + 1\right)\cdot 71^{7} + \left(50 a^{5} + 8 a^{4} + 43 a^{3} + 42 a^{2} + 21 a + 21\right)\cdot 71^{8} + \left(7 a^{5} + 5 a^{4} + 53 a^{3} + 32 a^{2} + 5 a + 29\right)\cdot 71^{9} +O(71^{10})$ 57a^5 + 63a^4 + 16a^3 + 31a^2 + 33a + 70 + (26a^5 + 68a^4 + 3a^3 + 43a^2 + 30a + 58)71 + (48a^5 + 69a^4 + 39a^3 + 61a^2 + 66a + 21)71^2 + (23a^5 + 62a^4 + 17a^3 + 15a^2 + 19a + 36)71^3 + (62a^5 + 66a^4 + 42a^3 + 47a^2 + 9a + 39)71^4 + (24a^5 + 27a^4 + 34a^3 + 35a^2 + 54a + 57)71^5 + (67a^5 + 69a^4 + 61a^3 + 2a^2 + 12a + 5)71^6 + (31a^5 + 65a^4 + 43a^3 + 34a^2 + 34a + 1)71^7 + (50a^5 + 8a^4 + 43a^3 + 42a^2 + 21a + 21)71^8 + (7a^5 + 5a^4 + 53a^3 + 32a^2 + 5a + 29)*71^9+O(71^10)
$r_{ 15 }$	$=$	$ 64 a^{5} + 47 a^{4} + 22 a^{3} + 3 a^{2} + 69 a + 3 + \left(21 a^{5} + 39 a^{4} + 16 a^{3} + 7 a^{2} + 38 a + 3\right)\cdot 71 + \left(2 a^{5} + 69 a^{4} + 44 a^{3} + 32 a^{2} + 19 a + 10\right)\cdot 71^{2} + \left(25 a^{5} + 23 a^{4} + 11 a^{3} + 24 a^{2} + 56 a + 53\right)\cdot 71^{3} + \left(47 a^{5} + 57 a^{4} + 23 a^{3} + 18 a^{2} + 58 a + 24\right)\cdot 71^{4} + \left(15 a^{5} + 33 a^{4} + 65 a^{3} + 29 a^{2} + 41 a + 4\right)\cdot 71^{5} + \left(68 a^{5} + 3 a^{4} + 37 a^{3} + 57 a^{2} + 61 a\right)\cdot 71^{6} + \left(44 a^{5} + 8 a^{4} + 43 a^{3} + 42 a^{2} + 17 a + 15\right)\cdot 71^{7} + \left(47 a^{5} + 34 a^{4} + a^{3} + 55 a^{2} + 62 a + 70\right)\cdot 71^{8} + \left(10 a^{5} + 35 a^{4} + 5 a^{3} + 24 a^{2} + 70 a + 60\right)\cdot 71^{9} +O(71^{10})$ 64a^5 + 47a^4 + 22a^3 + 3a^2 + 69a + 3 + (21a^5 + 39a^4 + 16a^3 + 7a^2 + 38a + 3)71 + (2a^5 + 69a^4 + 44a^3 + 32a^2 + 19a + 10)71^2 + (25a^5 + 23a^4 + 11a^3 + 24a^2 + 56a + 53)71^3 + (47a^5 + 57a^4 + 23a^3 + 18a^2 + 58a + 24)71^4 + (15a^5 + 33a^4 + 65a^3 + 29a^2 + 41a + 4)71^5 + (68a^5 + 3a^4 + 37a^3 + 57a^2 + 61a)71^6 + (44a^5 + 8a^4 + 43a^3 + 42a^2 + 17a + 15)71^7 + (47a^5 + 34a^4 + a^3 + 55a^2 + 62a + 70)71^8 + (10a^5 + 35a^4 + 5a^3 + 24a^2 + 70a + 60)71^9+O(71^10)
$r_{ 16 }$	$=$	$ 21 a^{5} + 49 a^{4} + 46 a^{3} + 50 a^{2} + 68 a + 13 + \left(23 a^{5} + 3 a^{4} + 49 a^{3} + 50 a^{2} + 20 a + 14\right)\cdot 71 + \left(2 a^{5} + 52 a^{4} + 63 a^{3} + 47 a^{2} + 11 a + 31\right)\cdot 71^{2} + \left(24 a^{5} + 18 a^{4} + 6 a^{3} + 66 a + 64\right)\cdot 71^{3} + \left(a^{5} + 41 a^{4} + 34 a^{3} + 68 a + 69\right)\cdot 71^{4} + \left(48 a^{5} + 45 a^{4} + 41 a^{3} + 47 a^{2} + 45 a + 52\right)\cdot 71^{5} + \left(4 a^{5} + 55 a^{4} + 42 a^{3} + 57 a^{2} + 50 a + 29\right)\cdot 71^{6} + \left(67 a^{5} + 27 a^{4} + 35 a^{3} + 37 a^{2} + 33 a + 9\right)\cdot 71^{7} + \left(61 a^{5} + 4 a^{4} + 47 a^{3} + 6 a^{2} + 46 a + 66\right)\cdot 71^{8} + \left(12 a^{5} + 45 a^{4} + 56 a^{3} + 15 a^{2} + 70 a + 50\right)\cdot 71^{9} +O(71^{10})$ 21a^5 + 49a^4 + 46a^3 + 50a^2 + 68a + 13 + (23a^5 + 3a^4 + 49a^3 + 50a^2 + 20a + 14)71 + (2a^5 + 52a^4 + 63a^3 + 47a^2 + 11a + 31)71^2 + (24a^5 + 18a^4 + 6a^3 + 66a + 64)71^3 + (a^5 + 41a^4 + 34a^3 + 68a + 69)71^4 + (48a^5 + 45a^4 + 41a^3 + 47a^2 + 45a + 52)71^5 + (4a^5 + 55a^4 + 42a^3 + 57a^2 + 50a + 29)71^6 + (67a^5 + 27a^4 + 35a^3 + 37a^2 + 33a + 9)71^7 + (61a^5 + 4a^4 + 47a^3 + 6a^2 + 46a + 66)71^8 + (12a^5 + 45a^4 + 56a^3 + 15a^2 + 70a + 50)71^9+O(71^10)

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

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 16 }$	Character values
			$c1$	$c2$	$c3$	$c4$
$1$	$1$	$()$	$2$	$2$	$2$	$2$
$1$	$2$	$(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$	$-2$	$-2$	$-2$	$-2$
$6$	$2$	$(1,7)(2,6)(3,4)(5,8)(9,15)(10,14)(11,12)(13,16)$	$0$	$0$	$0$	$0$
$4$	$3$	$(1,11,13)(3,5,9)(4,6,16)(8,12,14)$	$\zeta_{12}^{2}$	$-\zeta_{12}^{2} + 1$	$-\zeta_{12}^{2} + 1$	$\zeta_{12}^{2}$
$4$	$3$	$(1,13,11)(3,9,5)(4,16,6)(8,14,12)$	$-\zeta_{12}^{2} + 1$	$\zeta_{12}^{2}$	$\zeta_{12}^{2}$	$-\zeta_{12}^{2} + 1$
$1$	$4$	$(1,6,9,14)(2,15,10,7)(3,8,11,16)(4,5,12,13)$	$2 \zeta_{12}^{3}$	$-2 \zeta_{12}^{3}$	$2 \zeta_{12}^{3}$	$-2 \zeta_{12}^{3}$
$1$	$4$	$(1,14,9,6)(2,7,10,15)(3,16,11,8)(4,13,12,5)$	$-2 \zeta_{12}^{3}$	$2 \zeta_{12}^{3}$	$-2 \zeta_{12}^{3}$	$2 \zeta_{12}^{3}$
$6$	$4$	$(1,10,9,2)(3,13,11,5)(4,16,12,8)(6,7,14,15)$	$0$	$0$	$0$	$0$
$4$	$6$	$(1,5,11,9,13,3)(2,10)(4,8,6,12,16,14)(7,15)$	$\zeta_{12}^{2} - 1$	$-\zeta_{12}^{2}$	$-\zeta_{12}^{2}$	$\zeta_{12}^{2} - 1$
$4$	$6$	$(1,3,13,9,11,5)(2,10)(4,14,16,12,6,8)(7,15)$	$-\zeta_{12}^{2}$	$\zeta_{12}^{2} - 1$	$\zeta_{12}^{2} - 1$	$-\zeta_{12}^{2}$
$4$	$12$	$(1,16,5,14,11,4,9,8,13,6,3,12)(2,15,10,7)$	$\zeta_{12}^{3} - \zeta_{12}$	$-\zeta_{12}$	$\zeta_{12}$	$-\zeta_{12}^{3} + \zeta_{12}$
$4$	$12$	$(1,4,3,14,13,16,9,12,11,6,5,8)(2,15,10,7)$	$\zeta_{12}$	$-\zeta_{12}^{3} + \zeta_{12}$	$\zeta_{12}^{3} - \zeta_{12}$	$-\zeta_{12}$
$4$	$12$	$(1,8,5,6,11,12,9,16,13,14,3,4)(2,7,10,15)$	$-\zeta_{12}^{3} + \zeta_{12}$	$\zeta_{12}$	$-\zeta_{12}$	$\zeta_{12}^{3} - \zeta_{12}$
$4$	$12$	$(1,12,3,6,13,8,9,4,11,14,5,16)(2,7,10,15)$	$-\zeta_{12}$	$\zeta_{12}^{3} - \zeta_{12}$	$-\zeta_{12}^{3} + \zeta_{12}$	$\zeta_{12}$

The blue line marks the conjugacy class containing complex conjugation.

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Elliptic curves over $\Q$
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Basic invariants

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Roots of 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	2.403.16t60.a
Dimension	$2$
Group	$\SL(2,3):C_2$
Conductor	$403$
Indicator	$0$