LMFDB - Artin representation 2.1815.15t4.a.a

Basic invariants

Dimension:	$2$
Group:	$S_3 \times C_5$
Conductor:	$1815$$\medspace = 3 \cdot 5 \cdot 11^{2} $
Artin stem field:	Galois closure of 15.5.288372529877273634375.1
Galois orbit size:	$4$
Smallest permutation container:	$S_3 \times C_5$
Parity:	odd
Determinant:	1.165.10t1.a.b
Projective image:	$S_3$
Projective stem field:	Galois closure of 3.1.1815.1

Defining polynomial

$f(x)$

$=$

$ x^{15} - 3 x^{14} + 2 x^{13} - x^{12} - 17 x^{11} + 22 x^{10} + 99 x^{7} + 154 x^{6} + 132 x^{5} + \cdots + 197 $

The roots of $f$ are computed in an extension of $\Q_{ 19 }$ to precision 8.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 19 }$: $ x^{5} + 5x + 17 $ x^5 + 5*x + 17

Roots:

$r_{ 1 }$	$=$	$ 15 a^{3} + 8 a^{2} + 16 a + 4 + \left(10 a^{4} + 8 a^{3} + 8 a^{2} + 5 a + 17\right)\cdot 19 + \left(3 a^{4} + 17 a^{3} + 18 a^{2} + 5 a + 17\right)\cdot 19^{2} + \left(2 a^{4} + 2 a^{3} + 16 a^{2} + 8 a + 4\right)\cdot 19^{3} + \left(10 a^{4} + 9 a^{3} + 2 a^{2} + 12 a + 6\right)\cdot 19^{4} + \left(15 a^{4} + 6 a^{3} + 9 a^{2} + 7 a + 1\right)\cdot 19^{5} + \left(a^{4} + 6 a^{3} + 17 a^{2} + 9 a + 11\right)\cdot 19^{6} + \left(8 a^{4} + 5 a^{3} + 18 a + 9\right)\cdot 19^{7} +O(19^{8})$ 15a^3 + 8a^2 + 16a + 4 + (10a^4 + 8a^3 + 8a^2 + 5a + 17)19 + (3a^4 + 17a^3 + 18a^2 + 5a + 17)19^2 + (2a^4 + 2a^3 + 16a^2 + 8a + 4)19^3 + (10a^4 + 9a^3 + 2a^2 + 12a + 6)19^4 + (15a^4 + 6a^3 + 9a^2 + 7a + 1)19^5 + (a^4 + 6a^3 + 17a^2 + 9a + 11)19^6 + (8a^4 + 5a^3 + 18a + 9)19^7+O(19^8)
$r_{ 2 }$	$=$	$ 2 a^{4} + 10 a^{2} + 7 a + 12 + \left(11 a^{4} + 18 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 19 + \left(3 a^{4} + 11 a^{3} + 15 a^{2} + 11 a + 18\right)\cdot 19^{2} + \left(a^{3} + 6 a^{2} + 14 a + 15\right)\cdot 19^{3} + \left(11 a^{4} + 11 a^{2} + a + 9\right)\cdot 19^{4} + \left(10 a^{4} + 16 a^{3} + 6 a^{2} + 12 a\right)\cdot 19^{5} + \left(17 a^{4} + 6 a^{3} + 12 a^{2} + 3 a + 17\right)\cdot 19^{6} + \left(a^{4} + 6 a^{3} + 3 a + 3\right)\cdot 19^{7} +O(19^{8})$ 2a^4 + 10a^2 + 7a + 12 + (11a^4 + 18a^3 + 11a^2 + 12a + 2)19 + (3a^4 + 11a^3 + 15a^2 + 11a + 18)19^2 + (a^3 + 6a^2 + 14a + 15)19^3 + (11a^4 + 11a^2 + a + 9)19^4 + (10a^4 + 16a^3 + 6a^2 + 12a)19^5 + (17a^4 + 6a^3 + 12a^2 + 3a + 17)19^6 + (a^4 + 6a^3 + 3a + 3)19^7+O(19^8)
$r_{ 3 }$	$=$	$ 4 a^{4} + 13 a^{3} + 11 a^{2} + 2 a + 1 + \left(10 a^{4} + 14 a^{3} + 10 a^{2} + 16 a + 18\right)\cdot 19 + \left(9 a^{4} + 14 a^{3} + 6 a + 3\right)\cdot 19^{2} + \left(2 a^{4} + 4 a^{3} + 16 a^{2} + 5 a + 6\right)\cdot 19^{3} + \left(5 a^{4} + 16 a^{3} + 10 a^{2} + 17 a + 5\right)\cdot 19^{4} + \left(9 a^{4} + 14 a^{3} + 15 a^{2} + 7 a + 14\right)\cdot 19^{5} + \left(15 a^{4} + 8 a^{3} + 6 a^{2} + 9 a + 8\right)\cdot 19^{6} + \left(12 a^{4} + 15 a^{3} + 7 a^{2} + 9 a + 9\right)\cdot 19^{7} +O(19^{8})$ 4a^4 + 13a^3 + 11a^2 + 2a + 1 + (10a^4 + 14a^3 + 10a^2 + 16a + 18)19 + (9a^4 + 14a^3 + 6a + 3)19^2 + (2a^4 + 4a^3 + 16a^2 + 5a + 6)19^3 + (5a^4 + 16a^3 + 10a^2 + 17a + 5)19^4 + (9a^4 + 14a^3 + 15a^2 + 7a + 14)19^5 + (15a^4 + 8a^3 + 6a^2 + 9a + 8)19^6 + (12a^4 + 15a^3 + 7a^2 + 9a + 9)19^7+O(19^8)
$r_{ 4 }$	$=$	$ 5 a^{4} + 7 a^{3} + 15 a^{2} + 14 a + 5 + \left(13 a^{4} + 6 a^{3} + 3 a^{2} + 6 a + 11\right)\cdot 19 + \left(5 a^{4} + 2 a^{3} + a^{2} + 6 a + 7\right)\cdot 19^{2} + \left(7 a^{4} + 14 a^{3} + 4 a^{2} + 7 a + 6\right)\cdot 19^{3} + \left(a^{4} + 3 a^{3} + 5 a^{2} + 16 a + 9\right)\cdot 19^{4} + \left(10 a^{4} + 9 a^{3} + 4 a^{2} + 5 a + 17\right)\cdot 19^{5} + \left(2 a^{3} + 11 a^{2} + 5\right)\cdot 19^{6} + \left(13 a^{4} + 2 a^{2} + 3 a + 10\right)\cdot 19^{7} +O(19^{8})$ 5a^4 + 7a^3 + 15a^2 + 14a + 5 + (13a^4 + 6a^3 + 3a^2 + 6a + 11)19 + (5a^4 + 2a^3 + a^2 + 6a + 7)19^2 + (7a^4 + 14a^3 + 4a^2 + 7a + 6)19^3 + (a^4 + 3a^3 + 5a^2 + 16a + 9)19^4 + (10a^4 + 9a^3 + 4a^2 + 5a + 17)19^5 + (2a^3 + 11a^2 + 5)19^6 + (13a^4 + 2a^2 + 3a + 10)19^7+O(19^8)
$r_{ 5 }$	$=$	$ 6 a^{4} + 18 a^{3} + 6 a^{2} + 14 a + 9 + \left(3 a^{4} + 9 a^{2} + 17 a + 9\right)\cdot 19 + \left(15 a^{4} + 5 a^{3} + 3 a^{2} + 12 a + 7\right)\cdot 19^{2} + \left(8 a^{4} + 11 a^{3} + 11 a^{2} + 16 a + 12\right)\cdot 19^{3} + \left(16 a^{4} + 5 a^{3} + 13 a^{2} + 18 a + 12\right)\cdot 19^{4} + \left(11 a^{4} + 2 a^{2} + 18 a + 5\right)\cdot 19^{5} + \left(12 a^{4} + 6 a^{3} + 8 a^{2} + 14 a + 16\right)\cdot 19^{6} + \left(13 a^{4} + 15 a^{3} + 8 a^{2} + 8 a + 12\right)\cdot 19^{7} +O(19^{8})$ 6a^4 + 18a^3 + 6a^2 + 14a + 9 + (3a^4 + 9a^2 + 17a + 9)19 + (15a^4 + 5a^3 + 3a^2 + 12a + 7)19^2 + (8a^4 + 11a^3 + 11a^2 + 16a + 12)19^3 + (16a^4 + 5a^3 + 13a^2 + 18a + 12)19^4 + (11a^4 + 2a^2 + 18a + 5)19^5 + (12a^4 + 6a^3 + 8a^2 + 14a + 16)19^6 + (13a^4 + 15a^3 + 8a^2 + 8a + 12)*19^7+O(19^8)
$r_{ 6 }$	$=$	$ 7 a^{4} + 5 a^{2} + 9 a + 13 + \left(a^{4} + 18 a^{3} + 7 a^{2} + 16 a + 1\right)\cdot 19 + \left(6 a^{4} + 8 a^{2} + 18 a + 9\right)\cdot 19^{2} + \left(a^{4} + 13 a^{3} + 10 a^{2} + 6 a + 1\right)\cdot 19^{3} + \left(15 a^{4} + 9 a^{3} + 18 a^{2} + 7\right)\cdot 19^{4} + \left(17 a^{4} + 13 a^{3} + 4 a^{2} + 17 a + 10\right)\cdot 19^{5} + \left(10 a^{4} + 15 a^{3} + 18 a^{2} + 12 a + 9\right)\cdot 19^{6} + \left(15 a^{4} + 12 a^{3} + 13 a^{2} + 10 a + 1\right)\cdot 19^{7} +O(19^{8})$ 7a^4 + 5a^2 + 9a + 13 + (a^4 + 18a^3 + 7a^2 + 16a + 1)19 + (6a^4 + 8a^2 + 18a + 9)19^2 + (a^4 + 13a^3 + 10a^2 + 6a + 1)19^3 + (15a^4 + 9a^3 + 18a^2 + 7)19^4 + (17a^4 + 13a^3 + 4a^2 + 17a + 10)19^5 + (10a^4 + 15a^3 + 18a^2 + 12a + 9)19^6 + (15a^4 + 12a^3 + 13a^2 + 10a + 1)19^7+O(19^8)
$r_{ 7 }$	$=$	$ 7 a^{4} + 10 a^{3} + 17 a^{2} + 16 a + 13 + \left(14 a^{4} + 3 a^{3} + 12 a^{2} + 15 a + 15\right)\cdot 19 + \left(15 a^{4} + 13 a^{3} + 7 a^{2} + 11 a + 9\right)\cdot 19^{2} + \left(6 a^{4} + 6 a^{2} + 15 a + 4\right)\cdot 19^{3} + \left(17 a^{4} + 7 a^{3} + 6 a + 16\right)\cdot 19^{4} + \left(17 a^{4} + 15 a^{3} + 6 a^{2} + 7 a + 10\right)\cdot 19^{5} + \left(3 a^{4} + 14 a^{3} + 15 a^{2}\right)\cdot 19^{6} + \left(8 a^{4} + 10 a^{3} + 18 a^{2} + 15 a + 10\right)\cdot 19^{7} +O(19^{8})$ 7a^4 + 10a^3 + 17a^2 + 16a + 13 + (14a^4 + 3a^3 + 12a^2 + 15a + 15)19 + (15a^4 + 13a^3 + 7a^2 + 11a + 9)19^2 + (6a^4 + 6a^2 + 15a + 4)19^3 + (17a^4 + 7a^3 + 6a + 16)19^4 + (17a^4 + 15a^3 + 6a^2 + 7a + 10)19^5 + (3a^4 + 14a^3 + 15a^2)19^6 + (8a^4 + 10a^3 + 18a^2 + 15a + 10)19^7+O(19^8)
$r_{ 8 }$	$=$	$ 7 a^{4} + 12 a^{3} + 12 a^{2} + 13 + \left(a^{4} + 14 a^{3} + 4 a^{2} + 17 a + 1\right)\cdot 19 + \left(15 a^{4} + 15 a^{3} + 6 a + 7\right)\cdot 19^{2} + \left(15 a^{4} + 10 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 19^{3} + \left(8 a^{4} + 10 a^{3} + 17 a^{2} + 3 a + 1\right)\cdot 19^{4} + \left(2 a^{4} + 3 a^{3} + 12 a^{2} + 15 a + 6\right)\cdot 19^{5} + \left(15 a^{4} + a^{3} + 10 a^{2} + 15 a + 7\right)\cdot 19^{6} + \left(2 a^{4} + 16 a^{3} + 13 a^{2} + 8 a + 7\right)\cdot 19^{7} +O(19^{8})$ 7a^4 + 12a^3 + 12a^2 + 13 + (a^4 + 14a^3 + 4a^2 + 17a + 1)19 + (15a^4 + 15a^3 + 6a + 7)19^2 + (15a^4 + 10a^3 + 11a^2 + 12a + 2)19^3 + (8a^4 + 10a^3 + 17a^2 + 3a + 1)19^4 + (2a^4 + 3a^3 + 12a^2 + 15a + 6)19^5 + (15a^4 + a^3 + 10a^2 + 15a + 7)19^6 + (2a^4 + 16a^3 + 13a^2 + 8a + 7)*19^7+O(19^8)
$r_{ 9 }$	$=$	$ 7 a^{4} + 12 a^{3} + 17 a^{2} + 14 a + 13 + \left(12 a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 7\right)\cdot 19 + \left(10 a^{4} + a^{3} + 12 a^{2} + 16 a + 8\right)\cdot 19^{2} + \left(3 a^{4} + 3 a^{2} + 2 a + 10\right)\cdot 19^{3} + \left(16 a^{4} + 3 a^{3} + 16 a^{2} + 12 a + 11\right)\cdot 19^{4} + \left(17 a^{4} + 2 a^{2} + 14 a + 10\right)\cdot 19^{5} + \left(3 a^{4} + 13 a^{3} + 2 a^{2} + 8 a\right)\cdot 19^{6} + \left(15 a^{4} + 9 a^{3}\right)\cdot 19^{7} +O(19^{8})$ 7a^4 + 12a^3 + 17a^2 + 14a + 13 + (12a^4 + 3a^3 + 11a^2 + 16a + 7)19 + (10a^4 + a^3 + 12a^2 + 16a + 8)19^2 + (3a^4 + 3a^2 + 2a + 10)19^3 + (16a^4 + 3a^3 + 16a^2 + 12a + 11)19^4 + (17a^4 + 2a^2 + 14a + 10)19^5 + (3a^4 + 13a^3 + 2a^2 + 8a)19^6 + (15a^4 + 9a^3)19^7+O(19^8)
$r_{ 10 }$	$=$	$ 8 a^{4} + 6 a^{3} + 13 a^{2} + a + 17 + \left(13 a^{4} + 8 a^{3} + 12 a^{2} + 15 a + 11\right)\cdot 19 + \left(8 a^{4} + 4 a^{3} + 18 a^{2} + 8 a\right)\cdot 19^{2} + \left(4 a^{4} + 10 a^{3} + 18 a^{2} + 12 a + 14\right)\cdot 19^{3} + \left(14 a^{4} + 11 a^{3} + 18 a^{2} + 6 a + 3\right)\cdot 19^{4} + \left(7 a^{4} + 6 a^{3} + 5 a^{2} + 5 a + 8\right)\cdot 19^{5} + \left(10 a^{4} + 4 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 19^{6} + \left(7 a^{4} + 3 a^{3} + 14 a^{2} + 4 a + 7\right)\cdot 19^{7} +O(19^{8})$ 8a^4 + 6a^3 + 13a^2 + a + 17 + (13a^4 + 8a^3 + 12a^2 + 15a + 11)19 + (8a^4 + 4a^3 + 18a^2 + 8a)19^2 + (4a^4 + 10a^3 + 18a^2 + 12a + 14)19^3 + (14a^4 + 11a^3 + 18a^2 + 6a + 3)19^4 + (7a^4 + 6a^3 + 5a^2 + 5a + 8)19^5 + (10a^4 + 4a^3 + 5a^2 + 9a + 7)19^6 + (7a^4 + 3a^3 + 14a^2 + 4a + 7)19^7+O(19^8)
$r_{ 11 }$	$=$	$ 8 a^{4} + 15 a^{3} + 17 + \left(2 a^{4} + 3 a^{3} + 11 a + 5\right)\cdot 19 + \left(5 a^{4} + 16 a^{3} + 17 a^{2} + 16 a + 5\right)\cdot 19^{2} + \left(10 a^{4} + 12 a^{3} + 8 a^{2} + 4 a + 18\right)\cdot 19^{3} + \left(11 a^{4} + 10 a^{3} + 8 a^{2} + 14 a + 11\right)\cdot 19^{4} + \left(7 a^{4} + 8 a^{3} + 15 a^{2} + 8 a + 7\right)\cdot 19^{5} + \left(5 a^{4} + 17 a^{3} + 11 a^{2} + 13 a + 6\right)\cdot 19^{6} + \left(6 a^{4} + 5 a^{3} + a^{2} + 13 a + 2\right)\cdot 19^{7} +O(19^{8})$ 8a^4 + 15a^3 + 17 + (2a^4 + 3a^3 + 11a + 5)19 + (5a^4 + 16a^3 + 17a^2 + 16a + 5)19^2 + (10a^4 + 12a^3 + 8a^2 + 4a + 18)19^3 + (11a^4 + 10a^3 + 8a^2 + 14a + 11)19^4 + (7a^4 + 8a^3 + 15a^2 + 8a + 7)19^5 + (5a^4 + 17a^3 + 11a^2 + 13a + 6)19^6 + (6a^4 + 5a^3 + a^2 + 13a + 2)*19^7+O(19^8)
$r_{ 12 }$	$=$	$ 10 a^{4} + 12 a^{2} + 7 a + 6 + \left(13 a^{4} + 16 a^{3} + 8 a^{2} + 8 a + 12\right)\cdot 19 + \left(2 a^{4} + a^{3} + 12 a^{2} + 13 a + 14\right)\cdot 19^{2} + \left(9 a^{4} + 18 a^{2} + 16 a + 13\right)\cdot 19^{3} + \left(12 a^{4} + 5 a^{3} + 4 a^{2} + 12 a + 15\right)\cdot 19^{4} + \left(13 a^{4} + 17 a^{3} + 6 a^{2} + 10 a + 12\right)\cdot 19^{5} + \left(17 a^{4} + 17 a^{3} + 13 a^{2} + 14 a + 17\right)\cdot 19^{6} + \left(2 a^{4} + 17 a^{3} + a + 7\right)\cdot 19^{7} +O(19^{8})$ 10a^4 + 12a^2 + 7a + 6 + (13a^4 + 16a^3 + 8a^2 + 8a + 12)19 + (2a^4 + a^3 + 12a^2 + 13a + 14)19^2 + (9a^4 + 18a^2 + 16a + 13)19^3 + (12a^4 + 5a^3 + 4a^2 + 12a + 15)19^4 + (13a^4 + 17a^3 + 6a^2 + 10a + 12)19^5 + (17a^4 + 17a^3 + 13a^2 + 14a + 17)19^6 + (2a^4 + 17a^3 + a + 7)19^7+O(19^8)
$r_{ 13 }$	$=$	$ 11 a^{4} + 15 a^{3} + 17 a^{2} + a + 10 + \left(4 a^{2} + 8 a + 17\right)\cdot 19 + \left(10 a^{4} + 15 a^{3} + 13 a^{2} + a + 5\right)\cdot 19^{2} + \left(8 a^{4} + 2 a^{3} + a^{2} + 18 a + 11\right)\cdot 19^{3} + \left(9 a^{3} + 10 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(7 a^{4} + a^{3} + 14 a^{2} + 10 a + 5\right)\cdot 19^{5} + \left(14 a^{4} + 2 a^{3} + 17 a^{2} + 4 a + 4\right)\cdot 19^{6} + \left(9 a^{4} + 3 a^{3} + a^{2} + 9 a + 16\right)\cdot 19^{7} +O(19^{8})$ 11a^4 + 15a^3 + 17a^2 + a + 10 + (4a^2 + 8a + 17)19 + (10a^4 + 15a^3 + 13a^2 + a + 5)19^2 + (8a^4 + 2a^3 + a^2 + 18a + 11)19^3 + (9a^3 + 10a^2 + 2a + 5)19^4 + (7a^4 + a^3 + 14a^2 + 10a + 5)19^5 + (14a^4 + 2a^3 + 17a^2 + 4a + 4)19^6 + (9a^4 + 3a^3 + a^2 + 9a + 16)*19^7+O(19^8)
$r_{ 14 }$	$=$	$ 16 a^{4} + 3 a^{3} + 4 a^{2} + 16 a + 11 + \left(4 a^{4} + 15 a^{3} + 16 a^{2} + 11 a + 15\right)\cdot 19 + \left(2 a^{4} + 13 a^{3} + 15 a^{2} + 8 a + 12\right)\cdot 19^{2} + \left(16 a^{4} + 10 a^{3} + 16 a^{2} + 9 a + 3\right)\cdot 19^{3} + \left(13 a^{4} + 16 a^{3} + 15 a^{2} + 13 a + 2\right)\cdot 19^{4} + \left(13 a^{4} + 14 a^{2} + 14 a + 13\right)\cdot 19^{5} + \left(17 a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 17\right)\cdot 19^{6} + \left(5 a^{4} + 16 a^{3}\right)\cdot 19^{7} +O(19^{8})$ 16a^4 + 3a^3 + 4a^2 + 16a + 11 + (4a^4 + 15a^3 + 16a^2 + 11a + 15)19 + (2a^4 + 13a^3 + 15a^2 + 8a + 12)19^2 + (16a^4 + 10a^3 + 16a^2 + 9a + 3)19^3 + (13a^4 + 16a^3 + 15a^2 + 13a + 2)19^4 + (13a^4 + 14a^2 + 14a + 13)19^5 + (17a^4 + 3a^3 + 11a^2 + 16a + 17)19^6 + (5a^4 + 16a^3)19^7+O(19^8)
$r_{ 15 }$	$=$	$ 16 a^{4} + 7 a^{3} + 5 a^{2} + 16 a + 11 + \left(a^{4} + 10 a^{2} + 10 a + 3\right)\cdot 19 + \left(18 a^{3} + 6 a^{2} + 5 a + 4\right)\cdot 19^{2} + \left(17 a^{4} + 17 a^{3} + 7\right)\cdot 19^{3} + \left(16 a^{4} + 14 a^{3} + 16 a^{2} + 12 a + 14\right)\cdot 19^{4} + \left(7 a^{4} + 18 a^{3} + 10 a^{2} + 14 a + 8\right)\cdot 19^{5} + \left(4 a^{4} + 12 a^{3} + 8 a^{2} + 17 a + 2\right)\cdot 19^{6} + \left(9 a^{4} + 13 a^{3} + 9 a^{2} + 5 a + 14\right)\cdot 19^{7} +O(19^{8})$ 16a^4 + 7a^3 + 5a^2 + 16a + 11 + (a^4 + 10a^2 + 10a + 3)19 + (18a^3 + 6a^2 + 5a + 4)19^2 + (17a^4 + 17a^3 + 7)19^3 + (16a^4 + 14a^3 + 16a^2 + 12a + 14)19^4 + (7a^4 + 18a^3 + 10a^2 + 14a + 8)19^5 + (4a^4 + 12a^3 + 8a^2 + 17a + 2)19^6 + (9a^4 + 13a^3 + 9a^2 + 5a + 14)19^7+O(19^8)

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.

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Basic invariants

Defining polynomial

Generators of the action on the roots $r_1, \ldots, r_{ 15 }$

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	2.1815.15t4.a.a
Dimension	$2$
Group	$S_3 \times C_5$
Conductor	$1815$
Root number	not computed
Indicator	$0$

$r_{ 1 }$	$=$	\( 15 a^{3} + 8 a^{2} + 16 a + 4 + \left(10 a^{4} + 8 a^{3} + 8 a^{2} + 5 a + 17\right)\cdot 19 + \left(3 a^{4} + 17 a^{3} + 18 a^{2} + 5 a + 17\right)\cdot 19^{2} + \left(2 a^{4} + 2 a^{3} + 16 a^{2} + 8 a + 4\right)\cdot 19^{3} + \left(10 a^{4} + 9 a^{3} + 2 a^{2} + 12 a + 6\right)\cdot 19^{4} + \left(15 a^{4} + 6 a^{3} + 9 a^{2} + 7 a + 1\right)\cdot 19^{5} + \left(a^{4} + 6 a^{3} + 17 a^{2} + 9 a + 11\right)\cdot 19^{6} + \left(8 a^{4} + 5 a^{3} + 18 a + 9\right)\cdot 19^{7} +O(19^{8})\) 15a^3 + 8a^2 + 16a + 4 + (10a^4 + 8a^3 + 8a^2 + 5a + 17)19 + (3a^4 + 17a^3 + 18a^2 + 5a + 17)19^2 + (2a^4 + 2a^3 + 16a^2 + 8a + 4)19^3 + (10a^4 + 9a^3 + 2a^2 + 12a + 6)19^4 + (15a^4 + 6a^3 + 9a^2 + 7a + 1)19^5 + (a^4 + 6a^3 + 17a^2 + 9a + 11)19^6 + (8a^4 + 5a^3 + 18a + 9)19^7+O(19^8)
$r_{ 2 }$	$=$	\( 2 a^{4} + 10 a^{2} + 7 a + 12 + \left(11 a^{4} + 18 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 19 + \left(3 a^{4} + 11 a^{3} + 15 a^{2} + 11 a + 18\right)\cdot 19^{2} + \left(a^{3} + 6 a^{2} + 14 a + 15\right)\cdot 19^{3} + \left(11 a^{4} + 11 a^{2} + a + 9\right)\cdot 19^{4} + \left(10 a^{4} + 16 a^{3} + 6 a^{2} + 12 a\right)\cdot 19^{5} + \left(17 a^{4} + 6 a^{3} + 12 a^{2} + 3 a + 17\right)\cdot 19^{6} + \left(a^{4} + 6 a^{3} + 3 a + 3\right)\cdot 19^{7} +O(19^{8})\) 2a^4 + 10a^2 + 7a + 12 + (11a^4 + 18a^3 + 11a^2 + 12a + 2)19 + (3a^4 + 11a^3 + 15a^2 + 11a + 18)19^2 + (a^3 + 6a^2 + 14a + 15)19^3 + (11a^4 + 11a^2 + a + 9)19^4 + (10a^4 + 16a^3 + 6a^2 + 12a)19^5 + (17a^4 + 6a^3 + 12a^2 + 3a + 17)19^6 + (a^4 + 6a^3 + 3a + 3)19^7+O(19^8)
$r_{ 3 }$	$=$	\( 4 a^{4} + 13 a^{3} + 11 a^{2} + 2 a + 1 + \left(10 a^{4} + 14 a^{3} + 10 a^{2} + 16 a + 18\right)\cdot 19 + \left(9 a^{4} + 14 a^{3} + 6 a + 3\right)\cdot 19^{2} + \left(2 a^{4} + 4 a^{3} + 16 a^{2} + 5 a + 6\right)\cdot 19^{3} + \left(5 a^{4} + 16 a^{3} + 10 a^{2} + 17 a + 5\right)\cdot 19^{4} + \left(9 a^{4} + 14 a^{3} + 15 a^{2} + 7 a + 14\right)\cdot 19^{5} + \left(15 a^{4} + 8 a^{3} + 6 a^{2} + 9 a + 8\right)\cdot 19^{6} + \left(12 a^{4} + 15 a^{3} + 7 a^{2} + 9 a + 9\right)\cdot 19^{7} +O(19^{8})\) 4a^4 + 13a^3 + 11a^2 + 2a + 1 + (10a^4 + 14a^3 + 10a^2 + 16a + 18)19 + (9a^4 + 14a^3 + 6a + 3)19^2 + (2a^4 + 4a^3 + 16a^2 + 5a + 6)19^3 + (5a^4 + 16a^3 + 10a^2 + 17a + 5)19^4 + (9a^4 + 14a^3 + 15a^2 + 7a + 14)19^5 + (15a^4 + 8a^3 + 6a^2 + 9a + 8)19^6 + (12a^4 + 15a^3 + 7a^2 + 9a + 9)19^7+O(19^8)
$r_{ 4 }$	$=$	\( 5 a^{4} + 7 a^{3} + 15 a^{2} + 14 a + 5 + \left(13 a^{4} + 6 a^{3} + 3 a^{2} + 6 a + 11\right)\cdot 19 + \left(5 a^{4} + 2 a^{3} + a^{2} + 6 a + 7\right)\cdot 19^{2} + \left(7 a^{4} + 14 a^{3} + 4 a^{2} + 7 a + 6\right)\cdot 19^{3} + \left(a^{4} + 3 a^{3} + 5 a^{2} + 16 a + 9\right)\cdot 19^{4} + \left(10 a^{4} + 9 a^{3} + 4 a^{2} + 5 a + 17\right)\cdot 19^{5} + \left(2 a^{3} + 11 a^{2} + 5\right)\cdot 19^{6} + \left(13 a^{4} + 2 a^{2} + 3 a + 10\right)\cdot 19^{7} +O(19^{8})\) 5a^4 + 7a^3 + 15a^2 + 14a + 5 + (13a^4 + 6a^3 + 3a^2 + 6a + 11)19 + (5a^4 + 2a^3 + a^2 + 6a + 7)19^2 + (7a^4 + 14a^3 + 4a^2 + 7a + 6)19^3 + (a^4 + 3a^3 + 5a^2 + 16a + 9)19^4 + (10a^4 + 9a^3 + 4a^2 + 5a + 17)19^5 + (2a^3 + 11a^2 + 5)19^6 + (13a^4 + 2a^2 + 3a + 10)19^7+O(19^8)
$r_{ 5 }$	$=$	\( 6 a^{4} + 18 a^{3} + 6 a^{2} + 14 a + 9 + \left(3 a^{4} + 9 a^{2} + 17 a + 9\right)\cdot 19 + \left(15 a^{4} + 5 a^{3} + 3 a^{2} + 12 a + 7\right)\cdot 19^{2} + \left(8 a^{4} + 11 a^{3} + 11 a^{2} + 16 a + 12\right)\cdot 19^{3} + \left(16 a^{4} + 5 a^{3} + 13 a^{2} + 18 a + 12\right)\cdot 19^{4} + \left(11 a^{4} + 2 a^{2} + 18 a + 5\right)\cdot 19^{5} + \left(12 a^{4} + 6 a^{3} + 8 a^{2} + 14 a + 16\right)\cdot 19^{6} + \left(13 a^{4} + 15 a^{3} + 8 a^{2} + 8 a + 12\right)\cdot 19^{7} +O(19^{8})\) 6a^4 + 18a^3 + 6a^2 + 14a + 9 + (3a^4 + 9a^2 + 17a + 9)19 + (15a^4 + 5a^3 + 3a^2 + 12a + 7)19^2 + (8a^4 + 11a^3 + 11a^2 + 16a + 12)19^3 + (16a^4 + 5a^3 + 13a^2 + 18a + 12)19^4 + (11a^4 + 2a^2 + 18a + 5)19^5 + (12a^4 + 6a^3 + 8a^2 + 14a + 16)19^6 + (13a^4 + 15a^3 + 8a^2 + 8a + 12)*19^7+O(19^8)
$r_{ 6 }$	$=$	\( 7 a^{4} + 5 a^{2} + 9 a + 13 + \left(a^{4} + 18 a^{3} + 7 a^{2} + 16 a + 1\right)\cdot 19 + \left(6 a^{4} + 8 a^{2} + 18 a + 9\right)\cdot 19^{2} + \left(a^{4} + 13 a^{3} + 10 a^{2} + 6 a + 1\right)\cdot 19^{3} + \left(15 a^{4} + 9 a^{3} + 18 a^{2} + 7\right)\cdot 19^{4} + \left(17 a^{4} + 13 a^{3} + 4 a^{2} + 17 a + 10\right)\cdot 19^{5} + \left(10 a^{4} + 15 a^{3} + 18 a^{2} + 12 a + 9\right)\cdot 19^{6} + \left(15 a^{4} + 12 a^{3} + 13 a^{2} + 10 a + 1\right)\cdot 19^{7} +O(19^{8})\) 7a^4 + 5a^2 + 9a + 13 + (a^4 + 18a^3 + 7a^2 + 16a + 1)19 + (6a^4 + 8a^2 + 18a + 9)19^2 + (a^4 + 13a^3 + 10a^2 + 6a + 1)19^3 + (15a^4 + 9a^3 + 18a^2 + 7)19^4 + (17a^4 + 13a^3 + 4a^2 + 17a + 10)19^5 + (10a^4 + 15a^3 + 18a^2 + 12a + 9)19^6 + (15a^4 + 12a^3 + 13a^2 + 10a + 1)19^7+O(19^8)
$r_{ 7 }$	$=$	\( 7 a^{4} + 10 a^{3} + 17 a^{2} + 16 a + 13 + \left(14 a^{4} + 3 a^{3} + 12 a^{2} + 15 a + 15\right)\cdot 19 + \left(15 a^{4} + 13 a^{3} + 7 a^{2} + 11 a + 9\right)\cdot 19^{2} + \left(6 a^{4} + 6 a^{2} + 15 a + 4\right)\cdot 19^{3} + \left(17 a^{4} + 7 a^{3} + 6 a + 16\right)\cdot 19^{4} + \left(17 a^{4} + 15 a^{3} + 6 a^{2} + 7 a + 10\right)\cdot 19^{5} + \left(3 a^{4} + 14 a^{3} + 15 a^{2}\right)\cdot 19^{6} + \left(8 a^{4} + 10 a^{3} + 18 a^{2} + 15 a + 10\right)\cdot 19^{7} +O(19^{8})\) 7a^4 + 10a^3 + 17a^2 + 16a + 13 + (14a^4 + 3a^3 + 12a^2 + 15a + 15)19 + (15a^4 + 13a^3 + 7a^2 + 11a + 9)19^2 + (6a^4 + 6a^2 + 15a + 4)19^3 + (17a^4 + 7a^3 + 6a + 16)19^4 + (17a^4 + 15a^3 + 6a^2 + 7a + 10)19^5 + (3a^4 + 14a^3 + 15a^2)19^6 + (8a^4 + 10a^3 + 18a^2 + 15a + 10)19^7+O(19^8)
$r_{ 8 }$	$=$	\( 7 a^{4} + 12 a^{3} + 12 a^{2} + 13 + \left(a^{4} + 14 a^{3} + 4 a^{2} + 17 a + 1\right)\cdot 19 + \left(15 a^{4} + 15 a^{3} + 6 a + 7\right)\cdot 19^{2} + \left(15 a^{4} + 10 a^{3} + 11 a^{2} + 12 a + 2\right)\cdot 19^{3} + \left(8 a^{4} + 10 a^{3} + 17 a^{2} + 3 a + 1\right)\cdot 19^{4} + \left(2 a^{4} + 3 a^{3} + 12 a^{2} + 15 a + 6\right)\cdot 19^{5} + \left(15 a^{4} + a^{3} + 10 a^{2} + 15 a + 7\right)\cdot 19^{6} + \left(2 a^{4} + 16 a^{3} + 13 a^{2} + 8 a + 7\right)\cdot 19^{7} +O(19^{8})\) 7a^4 + 12a^3 + 12a^2 + 13 + (a^4 + 14a^3 + 4a^2 + 17a + 1)19 + (15a^4 + 15a^3 + 6a + 7)19^2 + (15a^4 + 10a^3 + 11a^2 + 12a + 2)19^3 + (8a^4 + 10a^3 + 17a^2 + 3a + 1)19^4 + (2a^4 + 3a^3 + 12a^2 + 15a + 6)19^5 + (15a^4 + a^3 + 10a^2 + 15a + 7)19^6 + (2a^4 + 16a^3 + 13a^2 + 8a + 7)*19^7+O(19^8)
$r_{ 9 }$	$=$	\( 7 a^{4} + 12 a^{3} + 17 a^{2} + 14 a + 13 + \left(12 a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 7\right)\cdot 19 + \left(10 a^{4} + a^{3} + 12 a^{2} + 16 a + 8\right)\cdot 19^{2} + \left(3 a^{4} + 3 a^{2} + 2 a + 10\right)\cdot 19^{3} + \left(16 a^{4} + 3 a^{3} + 16 a^{2} + 12 a + 11\right)\cdot 19^{4} + \left(17 a^{4} + 2 a^{2} + 14 a + 10\right)\cdot 19^{5} + \left(3 a^{4} + 13 a^{3} + 2 a^{2} + 8 a\right)\cdot 19^{6} + \left(15 a^{4} + 9 a^{3}\right)\cdot 19^{7} +O(19^{8})\) 7a^4 + 12a^3 + 17a^2 + 14a + 13 + (12a^4 + 3a^3 + 11a^2 + 16a + 7)19 + (10a^4 + a^3 + 12a^2 + 16a + 8)19^2 + (3a^4 + 3a^2 + 2a + 10)19^3 + (16a^4 + 3a^3 + 16a^2 + 12a + 11)19^4 + (17a^4 + 2a^2 + 14a + 10)19^5 + (3a^4 + 13a^3 + 2a^2 + 8a)19^6 + (15a^4 + 9a^3)19^7+O(19^8)
$r_{ 10 }$	$=$	\( 8 a^{4} + 6 a^{3} + 13 a^{2} + a + 17 + \left(13 a^{4} + 8 a^{3} + 12 a^{2} + 15 a + 11\right)\cdot 19 + \left(8 a^{4} + 4 a^{3} + 18 a^{2} + 8 a\right)\cdot 19^{2} + \left(4 a^{4} + 10 a^{3} + 18 a^{2} + 12 a + 14\right)\cdot 19^{3} + \left(14 a^{4} + 11 a^{3} + 18 a^{2} + 6 a + 3\right)\cdot 19^{4} + \left(7 a^{4} + 6 a^{3} + 5 a^{2} + 5 a + 8\right)\cdot 19^{5} + \left(10 a^{4} + 4 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 19^{6} + \left(7 a^{4} + 3 a^{3} + 14 a^{2} + 4 a + 7\right)\cdot 19^{7} +O(19^{8})\) 8a^4 + 6a^3 + 13a^2 + a + 17 + (13a^4 + 8a^3 + 12a^2 + 15a + 11)19 + (8a^4 + 4a^3 + 18a^2 + 8a)19^2 + (4a^4 + 10a^3 + 18a^2 + 12a + 14)19^3 + (14a^4 + 11a^3 + 18a^2 + 6a + 3)19^4 + (7a^4 + 6a^3 + 5a^2 + 5a + 8)19^5 + (10a^4 + 4a^3 + 5a^2 + 9a + 7)19^6 + (7a^4 + 3a^3 + 14a^2 + 4a + 7)19^7+O(19^8)
$r_{ 11 }$	$=$	\( 8 a^{4} + 15 a^{3} + 17 + \left(2 a^{4} + 3 a^{3} + 11 a + 5\right)\cdot 19 + \left(5 a^{4} + 16 a^{3} + 17 a^{2} + 16 a + 5\right)\cdot 19^{2} + \left(10 a^{4} + 12 a^{3} + 8 a^{2} + 4 a + 18\right)\cdot 19^{3} + \left(11 a^{4} + 10 a^{3} + 8 a^{2} + 14 a + 11\right)\cdot 19^{4} + \left(7 a^{4} + 8 a^{3} + 15 a^{2} + 8 a + 7\right)\cdot 19^{5} + \left(5 a^{4} + 17 a^{3} + 11 a^{2} + 13 a + 6\right)\cdot 19^{6} + \left(6 a^{4} + 5 a^{3} + a^{2} + 13 a + 2\right)\cdot 19^{7} +O(19^{8})\) 8a^4 + 15a^3 + 17 + (2a^4 + 3a^3 + 11a + 5)19 + (5a^4 + 16a^3 + 17a^2 + 16a + 5)19^2 + (10a^4 + 12a^3 + 8a^2 + 4a + 18)19^3 + (11a^4 + 10a^3 + 8a^2 + 14a + 11)19^4 + (7a^4 + 8a^3 + 15a^2 + 8a + 7)19^5 + (5a^4 + 17a^3 + 11a^2 + 13a + 6)19^6 + (6a^4 + 5a^3 + a^2 + 13a + 2)*19^7+O(19^8)
$r_{ 12 }$	$=$	\( 10 a^{4} + 12 a^{2} + 7 a + 6 + \left(13 a^{4} + 16 a^{3} + 8 a^{2} + 8 a + 12\right)\cdot 19 + \left(2 a^{4} + a^{3} + 12 a^{2} + 13 a + 14\right)\cdot 19^{2} + \left(9 a^{4} + 18 a^{2} + 16 a + 13\right)\cdot 19^{3} + \left(12 a^{4} + 5 a^{3} + 4 a^{2} + 12 a + 15\right)\cdot 19^{4} + \left(13 a^{4} + 17 a^{3} + 6 a^{2} + 10 a + 12\right)\cdot 19^{5} + \left(17 a^{4} + 17 a^{3} + 13 a^{2} + 14 a + 17\right)\cdot 19^{6} + \left(2 a^{4} + 17 a^{3} + a + 7\right)\cdot 19^{7} +O(19^{8})\) 10a^4 + 12a^2 + 7a + 6 + (13a^4 + 16a^3 + 8a^2 + 8a + 12)19 + (2a^4 + a^3 + 12a^2 + 13a + 14)19^2 + (9a^4 + 18a^2 + 16a + 13)19^3 + (12a^4 + 5a^3 + 4a^2 + 12a + 15)19^4 + (13a^4 + 17a^3 + 6a^2 + 10a + 12)19^5 + (17a^4 + 17a^3 + 13a^2 + 14a + 17)19^6 + (2a^4 + 17a^3 + a + 7)19^7+O(19^8)
$r_{ 13 }$	$=$	\( 11 a^{4} + 15 a^{3} + 17 a^{2} + a + 10 + \left(4 a^{2} + 8 a + 17\right)\cdot 19 + \left(10 a^{4} + 15 a^{3} + 13 a^{2} + a + 5\right)\cdot 19^{2} + \left(8 a^{4} + 2 a^{3} + a^{2} + 18 a + 11\right)\cdot 19^{3} + \left(9 a^{3} + 10 a^{2} + 2 a + 5\right)\cdot 19^{4} + \left(7 a^{4} + a^{3} + 14 a^{2} + 10 a + 5\right)\cdot 19^{5} + \left(14 a^{4} + 2 a^{3} + 17 a^{2} + 4 a + 4\right)\cdot 19^{6} + \left(9 a^{4} + 3 a^{3} + a^{2} + 9 a + 16\right)\cdot 19^{7} +O(19^{8})\) 11a^4 + 15a^3 + 17a^2 + a + 10 + (4a^2 + 8a + 17)19 + (10a^4 + 15a^3 + 13a^2 + a + 5)19^2 + (8a^4 + 2a^3 + a^2 + 18a + 11)19^3 + (9a^3 + 10a^2 + 2a + 5)19^4 + (7a^4 + a^3 + 14a^2 + 10a + 5)19^5 + (14a^4 + 2a^3 + 17a^2 + 4a + 4)19^6 + (9a^4 + 3a^3 + a^2 + 9a + 16)*19^7+O(19^8)
$r_{ 14 }$	$=$	\( 16 a^{4} + 3 a^{3} + 4 a^{2} + 16 a + 11 + \left(4 a^{4} + 15 a^{3} + 16 a^{2} + 11 a + 15\right)\cdot 19 + \left(2 a^{4} + 13 a^{3} + 15 a^{2} + 8 a + 12\right)\cdot 19^{2} + \left(16 a^{4} + 10 a^{3} + 16 a^{2} + 9 a + 3\right)\cdot 19^{3} + \left(13 a^{4} + 16 a^{3} + 15 a^{2} + 13 a + 2\right)\cdot 19^{4} + \left(13 a^{4} + 14 a^{2} + 14 a + 13\right)\cdot 19^{5} + \left(17 a^{4} + 3 a^{3} + 11 a^{2} + 16 a + 17\right)\cdot 19^{6} + \left(5 a^{4} + 16 a^{3}\right)\cdot 19^{7} +O(19^{8})\) 16a^4 + 3a^3 + 4a^2 + 16a + 11 + (4a^4 + 15a^3 + 16a^2 + 11a + 15)19 + (2a^4 + 13a^3 + 15a^2 + 8a + 12)19^2 + (16a^4 + 10a^3 + 16a^2 + 9a + 3)19^3 + (13a^4 + 16a^3 + 15a^2 + 13a + 2)19^4 + (13a^4 + 14a^2 + 14a + 13)19^5 + (17a^4 + 3a^3 + 11a^2 + 16a + 17)19^6 + (5a^4 + 16a^3)19^7+O(19^8)
$r_{ 15 }$	$=$	\( 16 a^{4} + 7 a^{3} + 5 a^{2} + 16 a + 11 + \left(a^{4} + 10 a^{2} + 10 a + 3\right)\cdot 19 + \left(18 a^{3} + 6 a^{2} + 5 a + 4\right)\cdot 19^{2} + \left(17 a^{4} + 17 a^{3} + 7\right)\cdot 19^{3} + \left(16 a^{4} + 14 a^{3} + 16 a^{2} + 12 a + 14\right)\cdot 19^{4} + \left(7 a^{4} + 18 a^{3} + 10 a^{2} + 14 a + 8\right)\cdot 19^{5} + \left(4 a^{4} + 12 a^{3} + 8 a^{2} + 17 a + 2\right)\cdot 19^{6} + \left(9 a^{4} + 13 a^{3} + 9 a^{2} + 5 a + 14\right)\cdot 19^{7} +O(19^{8})\) 16a^4 + 7a^3 + 5a^2 + 16a + 11 + (a^4 + 10a^2 + 10a + 3)19 + (18a^3 + 6a^2 + 5a + 4)19^2 + (17a^4 + 17a^3 + 7)19^3 + (16a^4 + 14a^3 + 16a^2 + 12a + 14)19^4 + (7a^4 + 18a^3 + 10a^2 + 14a + 8)19^5 + (4a^4 + 12a^3 + 8a^2 + 17a + 2)19^6 + (9a^4 + 13a^3 + 9a^2 + 5a + 14)19^7+O(19^8)