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Properties

Label	4.570025.10t11.a
Dimension	$4$
Group	$A_5\times C_2$
Conductor	$570025$
Indicator	$1$

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Underlying data

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Basic invariants

Dimension:	$4$
Group:	$A_5\times C_2$
Conductor:	\(570025\)\(\medspace = 5^{2} \cdot 151^{2} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 10.0.49064203594375.1
Galois orbit size:	$1$
Smallest permutation container:	$A_5\times C_2$
Parity:	even
Projective image:	$A_5$
Projective field:	Galois closure of 5.1.570025.1

Galois action

Roots of defining polynomial

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

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{5} + 7x + 28 \)

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Roots:

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

Generators of the action on the roots $r_1, \ldots, r_{ 10 }$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 10 }$	Character values
			$c1$
$1$	$1$	$()$	$4$
$1$	$2$	$(1,7)(2,5)(3,9)(4,8)(6,10)$	$-4$
$15$	$2$	$(1,4)(2,10)(3,9)(5,6)(7,8)$	$0$
$15$	$2$	$(1,8)(2,6)(4,7)(5,10)$	$0$
$20$	$3$	$(1,6,8)(4,7,10)$	$1$
$12$	$5$	$(1,3,8,2,6)(4,5,10,7,9)$	$-1$
$12$	$5$	$(1,2,3,6,8)(4,7,5,9,10)$	$-1$
$20$	$6$	$(1,4,6,7,8,10)(2,5)(3,9)$	$-1$
$12$	$10$	$(1,5,3,10,8,7,2,9,6,4)$	$1$
$12$	$10$	$(1,10,2,4,3,7,6,5,8,9)$	$1$

The blue line marks the conjugacy class containing complex conjugation.