LMFDB - Artin representation 2.1633.14t8.a.b

Basic invariants

Dimension:	$2$
Group:	$C_7 \wr C_2$
Conductor:	$1633$$\medspace = 23 \cdot 71 $
Artin stem field:	Galois closure of 14.0.436159106462145737687.1
Galois orbit size:	$6$
Smallest permutation container:	$C_7 \wr C_2$
Parity:	odd
Determinant:	1.1633.14t1.a.b
Projective image:	$D_7$
Projective stem field:	Galois closure of 7.1.1558596154466807.1

Defining polynomial

$f(x)$

$=$

$ x^{14} - x^{13} + 4 x^{12} + 2 x^{11} + 29 x^{10} + 61 x^{9} + 321 x^{8} + 535 x^{7} + 993 x^{6} + \cdots + 25 $

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: $ x^{7} + x + 28 $ x^7 + x + 28

Roots:

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

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

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

Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.

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Elliptic curves over $\Q$
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Basic invariants

Defining polynomial

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

Character values on conjugacy classes

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Label	2.1633.14t8.a.b
Dimension	$2$
Group	$C_7 \wr C_2$
Conductor	$1633$
Root number	not computed
Indicator	$0$