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Properties

Label	11.147...000.12t272.a
Dimension	$11$
Group	$M_{11}$
Conductor	$1.476\times 10^{24}$
Indicator	$1$

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Underlying data

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Basic invariants

Dimension:	$11$
Group:	$M_{11}$
Conductor:	\(147\!\cdots\!000\)\(\medspace = 2^{18} \cdot 3^{10} \cdot 5^{20} \)
Frobenius-Schur indicator:	$1$
Root number:	$1$
Artin number field:	Galois closure of 11.3.6561000000000000000000.1
Galois orbit size:	$1$
Smallest permutation container:	$M_{11}$
Parity:	even
Projective image:	$M_{11}$
Projective field:	Galois closure of 11.3.6561000000000000000000.1

Galois action

Roots of defining polynomial

The roots of $f$ are computed in an extension of $\Q_{ 13 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 13 }$: \( x^{5} + 4x + 11 \)

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Roots:

$r_{ 1 }$	$=$	\( 7 a^{4} + 3 a^{3} + 9 a^{2} + 6 a + 6 + \left(a^{4} + 6 a^{3} + 10 a^{2} + 6 a + 7\right)\cdot 13 + \left(11 a^{4} + 7 a^{3} + 2 a^{2} + 11 a + 10\right)\cdot 13^{2} + \left(6 a^{4} + 10 a^{3} + 12 a^{2} + 8 a + 12\right)\cdot 13^{3} + \left(4 a^{4} + 10 a^{3} + 8 a^{2} + 5\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + a^{2} + 7 a + 12\right)\cdot 13^{5} + \left(11 a^{4} + 11 a^{3} + 9 a^{2} + 4\right)\cdot 13^{6} + \left(2 a^{4} + a^{3} + 9 a^{2} + 7 a + 8\right)\cdot 13^{7} + \left(12 a^{4} + 2 a^{3} + 6 a^{2} + a + 8\right)\cdot 13^{8} + \left(2 a^{4} + 2 a^{3} + 3 a^{2} + 10 a + 2\right)\cdot 13^{9} +O(13^{10})\) 7a^4 + 3a^3 + 9a^2 + 6a + 6 + (a^4 + 6a^3 + 10a^2 + 6a + 7)13 + (11a^4 + 7a^3 + 2a^2 + 11a + 10)13^2 + (6a^4 + 10a^3 + 12a^2 + 8a + 12)13^3 + (4a^4 + 10a^3 + 8a^2 + 5)13^4 + (2a^4 + 6a^3 + a^2 + 7a + 12)13^5 + (11a^4 + 11a^3 + 9a^2 + 4)13^6 + (2a^4 + a^3 + 9a^2 + 7a + 8)13^7 + (12a^4 + 2a^3 + 6a^2 + a + 8)13^8 + (2a^4 + 2a^3 + 3a^2 + 10a + 2)*13^9+O(13^10)
$r_{ 2 }$	$=$	\( a^{4} + 2 a^{3} + 9 a^{2} + 4 a + 3 + \left(5 a^{4} + 3 a^{3} + 7 a^{2} + 6 a + 8\right)\cdot 13 + \left(6 a^{4} + 8 a^{3} + 9 a^{2} + a\right)\cdot 13^{2} + \left(3 a^{4} + 10 a^{3} + 12 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(10 a^{4} + 3 a^{3} + 2\right)\cdot 13^{4} + \left(11 a^{3} + 11 a^{2} + 5 a + 9\right)\cdot 13^{5} + \left(8 a^{4} + 4 a^{3} + a^{2} + 2 a + 2\right)\cdot 13^{6} + \left(5 a^{4} + 7 a^{3} + a + 5\right)\cdot 13^{7} + \left(12 a^{4} + 11 a^{3} + a^{2} + 4 a + 6\right)\cdot 13^{8} + \left(10 a^{4} + a^{3} + 6 a^{2} + 2 a\right)\cdot 13^{9} +O(13^{10})\) a^4 + 2a^3 + 9a^2 + 4a + 3 + (5a^4 + 3a^3 + 7a^2 + 6a + 8)13 + (6a^4 + 8a^3 + 9a^2 + a)13^2 + (3a^4 + 10a^3 + 12a^2 + 11a + 10)13^3 + (10a^4 + 3a^3 + 2)13^4 + (11a^3 + 11a^2 + 5a + 9)13^5 + (8a^4 + 4a^3 + a^2 + 2a + 2)13^6 + (5a^4 + 7a^3 + a + 5)13^7 + (12a^4 + 11a^3 + a^2 + 4a + 6)13^8 + (10a^4 + a^3 + 6a^2 + 2a)*13^9+O(13^10)
$r_{ 3 }$	$=$	\( 3 a^{4} + 6 a^{3} + 5 a^{2} + 12 a + 1 + \left(12 a^{4} + 6 a^{3} + 6 a^{2} + 11 a\right)\cdot 13 + \left(11 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 13^{2} + \left(6 a^{4} + 9 a^{3} + 7 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(12 a^{4} + a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 13^{4} + \left(9 a^{4} + a^{3} + 4 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(4 a^{4} + 6 a^{3} + 10 a + 10\right)\cdot 13^{6} + \left(8 a^{4} + 9 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{7} + \left(4 a^{4} + 8 a^{3} + 3 a^{2} + 2 a + 2\right)\cdot 13^{8} + \left(7 a^{4} + a^{3} + 7 a^{2} + 10 a + 1\right)\cdot 13^{9} +O(13^{10})\) 3a^4 + 6a^3 + 5a^2 + 12a + 1 + (12a^4 + 6a^3 + 6a^2 + 11a)13 + (11a^4 + 10a^3 + 9a^2 + 8a + 8)13^2 + (6a^4 + 9a^3 + 7a^2 + 11a + 2)13^3 + (12a^4 + a^3 + 6a^2 + 3a + 8)13^4 + (9a^4 + a^3 + 4a^2 + 7a + 5)13^5 + (4a^4 + 6a^3 + 10a + 10)13^6 + (8a^4 + 9a^3 + 5a^2 + 9a + 7)13^7 + (4a^4 + 8a^3 + 3a^2 + 2a + 2)13^8 + (7a^4 + a^3 + 7a^2 + 10a + 1)*13^9+O(13^10)
$r_{ 4 }$	$=$	\( a^{4} + 8 a^{3} + 4 a^{2} + 7 a + 3 + \left(8 a^{4} + 7 a^{3} + 4 a^{2} + 12 a + 2\right)\cdot 13 + \left(4 a^{4} + 8 a^{3} + 7 a^{2} + 8 a\right)\cdot 13^{2} + \left(2 a^{4} + 4 a^{3} + 3 a + 9\right)\cdot 13^{3} + \left(11 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + 10 a + 5\right)\cdot 13^{5} + \left(6 a^{4} + 6 a^{3} + 7 a + 12\right)\cdot 13^{6} + \left(7 a^{4} + 10 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 13^{7} + \left(a^{4} + 5 a^{3} + 8 a^{2} + 2 a + 5\right)\cdot 13^{8} + \left(8 a^{4} + 12 a^{3} + 10 a^{2} + 11 a + 4\right)\cdot 13^{9} +O(13^{10})\) a^4 + 8a^3 + 4a^2 + 7a + 3 + (8a^4 + 7a^3 + 4a^2 + 12a + 2)13 + (4a^4 + 8a^3 + 7a^2 + 8a)13^2 + (2a^4 + 4a^3 + 3a + 9)13^3 + (11a^4 + 10a^3 + 10a^2 + 4a + 5)13^4 + (2a^4 + 6a^3 + 10a + 5)13^5 + (6a^4 + 6a^3 + 7a + 12)13^6 + (7a^4 + 10a^3 + 10a^2 + 8a + 5)13^7 + (a^4 + 5a^3 + 8a^2 + 2a + 5)13^8 + (8a^4 + 12a^3 + 10a^2 + 11a + 4)13^9+O(13^10)
$r_{ 5 }$	$=$	\( 4 a^{3} + 3 a^{2} + 4 a + 7 + \left(5 a^{4} + 3 a^{3} + 2 a^{2} + 5\right)\cdot 13 + \left(7 a^{4} + 9 a^{3} + 4 a + 6\right)\cdot 13^{2} + \left(9 a^{4} + 7 a^{2} + 12 a + 8\right)\cdot 13^{3} + \left(5 a^{4} + 4 a^{2} + 2 a + 4\right)\cdot 13^{4} + \left(5 a^{4} + 5 a^{3} + 6 a^{2} + a + 9\right)\cdot 13^{5} + \left(7 a^{3} + 8 a^{2} + 12 a + 1\right)\cdot 13^{6} + \left(9 a^{4} + a^{3} + 3 a^{2} + 7 a + 2\right)\cdot 13^{7} + \left(5 a^{4} + 12 a^{3} + 8 a^{2} + 6 a + 11\right)\cdot 13^{8} + \left(11 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 3\right)\cdot 13^{9} +O(13^{10})\) 4a^3 + 3a^2 + 4a + 7 + (5a^4 + 3a^3 + 2a^2 + 5)13 + (7a^4 + 9a^3 + 4a + 6)13^2 + (9a^4 + 7a^2 + 12a + 8)13^3 + (5a^4 + 4a^2 + 2a + 4)13^4 + (5a^4 + 5a^3 + 6a^2 + a + 9)13^5 + (7a^3 + 8a^2 + 12a + 1)13^6 + (9a^4 + a^3 + 3a^2 + 7a + 2)13^7 + (5a^4 + 12a^3 + 8a^2 + 6a + 11)13^8 + (11a^4 + 10a^3 + 9a^2 + 8a + 3)*13^9+O(13^10)
$r_{ 6 }$	$=$	\( 6 a^{4} + a^{3} + 8 a^{2} + 6 a + 8 + \left(9 a^{4} + 5 a^{3} + 6 a^{2} + 5 a + 1\right)\cdot 13 + \left(9 a^{4} + 6 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 13^{2} + \left(9 a^{4} + 3 a^{3} + 11 a^{2} + 10 a + 6\right)\cdot 13^{3} + \left(5 a^{4} + 9 a^{3} + 12\right)\cdot 13^{4} + \left(8 a^{4} + 12 a^{3} + 9 a^{2} + 2 a + 5\right)\cdot 13^{5} + \left(9 a^{4} + 6 a^{3} + 5 a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(12 a^{4} + 2 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 13^{7} + \left(6 a^{4} + 12 a^{3} + 8 a^{2} + a + 12\right)\cdot 13^{8} + \left(8 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{9} +O(13^{10})\) 6a^4 + a^3 + 8a^2 + 6a + 8 + (9a^4 + 5a^3 + 6a^2 + 5a + 1)13 + (9a^4 + 6a^3 + 6a^2 + 12a + 6)13^2 + (9a^4 + 3a^3 + 11a^2 + 10a + 6)13^3 + (5a^4 + 9a^3 + 12)13^4 + (8a^4 + 12a^3 + 9a^2 + 2a + 5)13^5 + (9a^4 + 6a^3 + 5a^2 + 7a + 10)13^6 + (12a^4 + 2a^3 + 3a^2 + 5a + 8)13^7 + (6a^4 + 12a^3 + 8a^2 + a + 12)13^8 + (8a^3 + 5a^2 + 9a + 7)13^9+O(13^10)
$r_{ 7 }$	$=$	\( 7 + 13 + 3\cdot 13^{2} + 12\cdot 13^{3} + 10\cdot 13^{4} + 6\cdot 13^{5} + 9\cdot 13^{6} + 2\cdot 13^{7} + 7\cdot 13^{8} + 11\cdot 13^{9} +O(13^{10})\) 7 + 13 + 313^2 + 1213^3 + 1013^4 + 613^5 + 913^6 + 213^7 + 713^8 + 1113^9+O(13^10)
$r_{ 8 }$	$=$	\( 10 a^{4} + 12 a^{3} + a^{2} + 11 a + \left(10 a^{4} + 4 a^{3} + a + 11\right)\cdot 13 + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + 2 a + 12\right)\cdot 13^{2} + \left(5 a^{4} + a^{3} + 8 a + 1\right)\cdot 13^{3} + \left(10 a^{4} + 4 a^{3} + 5 a^{2} + 4 a + 4\right)\cdot 13^{4} + \left(12 a^{4} + 4 a^{2} + 8 a + 4\right)\cdot 13^{5} + \left(12 a^{4} + 7 a^{3} + 2 a^{2} + 8 a\right)\cdot 13^{6} + \left(5 a^{4} + 10 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(9 a^{4} + 3 a^{3} + 12 a^{2} + 7\right)\cdot 13^{8} + \left(3 a^{4} + 2 a^{3} + 12 a^{2} + a + 2\right)\cdot 13^{9} +O(13^{10})\) 10a^4 + 12a^3 + a^2 + 11a + (10a^4 + 4a^3 + a + 11)13 + (11a^4 + 5a^3 + 7a^2 + 2a + 12)13^2 + (5a^4 + a^3 + 8a + 1)13^3 + (10a^4 + 4a^3 + 5a^2 + 4a + 4)13^4 + (12a^4 + 4a^2 + 8a + 4)13^5 + (12a^4 + 7a^3 + 2a^2 + 8a)13^6 + (5a^4 + 10a^3 + 4a^2 + 8a + 8)13^7 + (9a^4 + 3a^3 + 12a^2 + 7)13^8 + (3a^4 + 2a^3 + 12a^2 + a + 2)*13^9+O(13^10)
$r_{ 9 }$	$=$	\( 3 a^{4} + 6 a^{3} + 2 a^{2} + a + 12 + \left(12 a^{4} + 7 a^{3} + 12 a^{2} + 4 a + 12\right)\cdot 13 + \left(2 a^{4} + 3 a^{3} + 9 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(4 a^{4} + 6 a^{3} + 4 a^{2} + 2 a + 1\right)\cdot 13^{3} + \left(3 a^{4} + 7 a^{3} + 5 a^{2} + 11 a + 9\right)\cdot 13^{4} + \left(10 a^{4} + 2 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(4 a^{4} + 5 a^{3} + a^{2} + 8 a + 10\right)\cdot 13^{6} + \left(2 a^{4} + 6 a^{3} + 2 a^{2} + 8 a + 7\right)\cdot 13^{7} + \left(2 a^{4} + 3 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{8} + \left(11 a^{4} + 6 a^{3} + 10 a^{2} + 8\right)\cdot 13^{9} +O(13^{10})\) 3a^4 + 6a^3 + 2a^2 + a + 12 + (12a^4 + 7a^3 + 12a^2 + 4a + 12)13 + (2a^4 + 3a^3 + 9a^2 + 12a + 12)13^2 + (4a^4 + 6a^3 + 4a^2 + 2a + 1)13^3 + (3a^4 + 7a^3 + 5a^2 + 11a + 9)13^4 + (10a^4 + 2a^3 + 6a^2 + 7a + 5)13^5 + (4a^4 + 5a^3 + a^2 + 8a + 10)13^6 + (2a^4 + 6a^3 + 2a^2 + 8a + 7)13^7 + (2a^4 + 3a^3 + 3a^2 + 10a + 12)13^8 + (11a^4 + 6a^3 + 10a^2 + 8)13^9+O(13^10)
$r_{ 10 }$	$=$	\( 2 a^{4} + 9 a^{3} + a^{2} + 4 a + 1 + \left(11 a^{4} + 2 a^{3} + 9 a^{2} + 11 a + 12\right)\cdot 13 + \left(10 a^{4} + 8 a^{3} + 3 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(12 a^{4} + 2 a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(8 a^{4} + 5 a^{3} + 10 a + 6\right)\cdot 13^{4} + \left(2 a^{4} + 5 a^{3} + 12 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(11 a^{4} + 5 a^{3} + 8 a^{2} + 5 a + 7\right)\cdot 13^{6} + \left(a^{4} + 11 a^{2} + 11 a + 3\right)\cdot 13^{7} + \left(6 a^{4} + 7 a^{3} + 10 a^{2} + a + 12\right)\cdot 13^{8} + \left(12 a^{4} + 4 a^{2} + a + 7\right)\cdot 13^{9} +O(13^{10})\) 2a^4 + 9a^3 + a^2 + 4a + 1 + (11a^4 + 2a^3 + 9a^2 + 11a + 12)13 + (10a^4 + 8a^3 + 3a^2 + 5a + 1)13^2 + (12a^4 + 2a^2 + 5a + 6)13^3 + (8a^4 + 5a^3 + 10a + 6)13^4 + (2a^4 + 5a^3 + 12a^2 + 2a + 7)13^5 + (11a^4 + 5a^3 + 8a^2 + 5a + 7)13^6 + (a^4 + 11a^2 + 11a + 3)13^7 + (6a^4 + 7a^3 + 10a^2 + a + 12)13^8 + (12a^4 + 4a^2 + a + 7)*13^9+O(13^10)
$r_{ 11 }$	$=$	\( 6 a^{4} + a^{3} + 10 a^{2} + 10 a + 6 + \left(2 a^{4} + 5 a^{3} + 5 a^{2} + 4 a + 2\right)\cdot 13 + \left(a^{4} + 10 a^{3} + 8 a^{2} + 10 a + 2\right)\cdot 13^{2} + \left(3 a^{4} + 3 a^{3} + 5 a^{2} + 2 a + 6\right)\cdot 13^{3} + \left(5 a^{4} + 12 a^{3} + 9 a^{2} + 12 a + 7\right)\cdot 13^{4} + \left(9 a^{4} + 12 a^{3} + 8 a^{2} + 12 a + 5\right)\cdot 13^{5} + \left(8 a^{4} + 3 a^{3} + a + 7\right)\cdot 13^{6} + \left(8 a^{4} + a^{3} + 2 a^{2} + 9 a + 4\right)\cdot 13^{7} + \left(3 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 4\right)\cdot 13^{8} + \left(9 a^{4} + 4 a^{3} + 7 a^{2} + 10 a\right)\cdot 13^{9} +O(13^{10})\) 6a^4 + a^3 + 10a^2 + 10a + 6 + (2a^4 + 5a^3 + 5a^2 + 4a + 2)13 + (a^4 + 10a^3 + 8a^2 + 10a + 2)13^2 + (3a^4 + 3a^3 + 5a^2 + 2a + 6)13^3 + (5a^4 + 12a^3 + 9a^2 + 12a + 7)13^4 + (9a^4 + 12a^3 + 8a^2 + 12a + 5)13^5 + (8a^4 + 3a^3 + a + 7)13^6 + (8a^4 + a^3 + 2a^2 + 9a + 4)13^7 + (3a^4 + 11a^3 + 2a^2 + 6a + 4)13^8 + (9a^4 + 4a^3 + 7a^2 + 10a)13^9+O(13^10)

Generators of the action on the roots $r_1, \ldots, r_{ 11 }$

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

Character values on conjugacy classes

Size	Order	Action on $r_1, \ldots, r_{ 11 }$	Character values
			$c1$
$1$	$1$	$()$	$11$
$165$	$2$	$(3,8)(4,6)(5,10)(7,11)$	$3$
$440$	$3$	$(2,9,10)(3,6,8)(4,11,7)$	$2$
$990$	$4$	$(3,7,8,11)(4,5,6,10)$	$-1$
$1584$	$5$	$(1,3,6,2,11)(4,10,5,8,7)$	$1$
$1320$	$6$	$(1,5,3)(2,8,10,4,6,11)(7,9)$	$0$
$990$	$8$	$(2,9)(3,10,7,4,8,5,11,6)$	$-1$
$990$	$8$	$(2,9)(3,5,7,6,8,10,11,4)$	$-1$
$720$	$11$	$(1,6,8,4,3,10,2,5,7,11,9)$	$0$
$720$	$11$	$(1,8,3,2,7,9,6,4,10,5,11)$	$0$

The blue line marks the conjugacy class containing complex conjugation.