Properties

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

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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 $\mathbb{Q}_{ 13 }$: \( x^{5} + 4x + 11 \) Copy content Toggle raw display
Roots:
$r_{ 1 }$ $=$ \( 7 a^{4} + 3 a^{3} + 9 a^{2} + 6 a + 6 + \left(a^{4} + 6 a^{3} + 10 a^{2} + 6 a + 7\right)\cdot 13 + \left(11 a^{4} + 7 a^{3} + 2 a^{2} + 11 a + 10\right)\cdot 13^{2} + \left(6 a^{4} + 10 a^{3} + 12 a^{2} + 8 a + 12\right)\cdot 13^{3} + \left(4 a^{4} + 10 a^{3} + 8 a^{2} + 5\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + a^{2} + 7 a + 12\right)\cdot 13^{5} + \left(11 a^{4} + 11 a^{3} + 9 a^{2} + 4\right)\cdot 13^{6} + \left(2 a^{4} + a^{3} + 9 a^{2} + 7 a + 8\right)\cdot 13^{7} + \left(12 a^{4} + 2 a^{3} + 6 a^{2} + a + 8\right)\cdot 13^{8} + \left(2 a^{4} + 2 a^{3} + 3 a^{2} + 10 a + 2\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( a^{4} + 2 a^{3} + 9 a^{2} + 4 a + 3 + \left(5 a^{4} + 3 a^{3} + 7 a^{2} + 6 a + 8\right)\cdot 13 + \left(6 a^{4} + 8 a^{3} + 9 a^{2} + a\right)\cdot 13^{2} + \left(3 a^{4} + 10 a^{3} + 12 a^{2} + 11 a + 10\right)\cdot 13^{3} + \left(10 a^{4} + 3 a^{3} + 2\right)\cdot 13^{4} + \left(11 a^{3} + 11 a^{2} + 5 a + 9\right)\cdot 13^{5} + \left(8 a^{4} + 4 a^{3} + a^{2} + 2 a + 2\right)\cdot 13^{6} + \left(5 a^{4} + 7 a^{3} + a + 5\right)\cdot 13^{7} + \left(12 a^{4} + 11 a^{3} + a^{2} + 4 a + 6\right)\cdot 13^{8} + \left(10 a^{4} + a^{3} + 6 a^{2} + 2 a\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a^{4} + 6 a^{3} + 5 a^{2} + 12 a + 1 + \left(12 a^{4} + 6 a^{3} + 6 a^{2} + 11 a\right)\cdot 13 + \left(11 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 8\right)\cdot 13^{2} + \left(6 a^{4} + 9 a^{3} + 7 a^{2} + 11 a + 2\right)\cdot 13^{3} + \left(12 a^{4} + a^{3} + 6 a^{2} + 3 a + 8\right)\cdot 13^{4} + \left(9 a^{4} + a^{3} + 4 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(4 a^{4} + 6 a^{3} + 10 a + 10\right)\cdot 13^{6} + \left(8 a^{4} + 9 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{7} + \left(4 a^{4} + 8 a^{3} + 3 a^{2} + 2 a + 2\right)\cdot 13^{8} + \left(7 a^{4} + a^{3} + 7 a^{2} + 10 a + 1\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( a^{4} + 8 a^{3} + 4 a^{2} + 7 a + 3 + \left(8 a^{4} + 7 a^{3} + 4 a^{2} + 12 a + 2\right)\cdot 13 + \left(4 a^{4} + 8 a^{3} + 7 a^{2} + 8 a\right)\cdot 13^{2} + \left(2 a^{4} + 4 a^{3} + 3 a + 9\right)\cdot 13^{3} + \left(11 a^{4} + 10 a^{3} + 10 a^{2} + 4 a + 5\right)\cdot 13^{4} + \left(2 a^{4} + 6 a^{3} + 10 a + 5\right)\cdot 13^{5} + \left(6 a^{4} + 6 a^{3} + 7 a + 12\right)\cdot 13^{6} + \left(7 a^{4} + 10 a^{3} + 10 a^{2} + 8 a + 5\right)\cdot 13^{7} + \left(a^{4} + 5 a^{3} + 8 a^{2} + 2 a + 5\right)\cdot 13^{8} + \left(8 a^{4} + 12 a^{3} + 10 a^{2} + 11 a + 4\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 4 a^{3} + 3 a^{2} + 4 a + 7 + \left(5 a^{4} + 3 a^{3} + 2 a^{2} + 5\right)\cdot 13 + \left(7 a^{4} + 9 a^{3} + 4 a + 6\right)\cdot 13^{2} + \left(9 a^{4} + 7 a^{2} + 12 a + 8\right)\cdot 13^{3} + \left(5 a^{4} + 4 a^{2} + 2 a + 4\right)\cdot 13^{4} + \left(5 a^{4} + 5 a^{3} + 6 a^{2} + a + 9\right)\cdot 13^{5} + \left(7 a^{3} + 8 a^{2} + 12 a + 1\right)\cdot 13^{6} + \left(9 a^{4} + a^{3} + 3 a^{2} + 7 a + 2\right)\cdot 13^{7} + \left(5 a^{4} + 12 a^{3} + 8 a^{2} + 6 a + 11\right)\cdot 13^{8} + \left(11 a^{4} + 10 a^{3} + 9 a^{2} + 8 a + 3\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( 6 a^{4} + a^{3} + 8 a^{2} + 6 a + 8 + \left(9 a^{4} + 5 a^{3} + 6 a^{2} + 5 a + 1\right)\cdot 13 + \left(9 a^{4} + 6 a^{3} + 6 a^{2} + 12 a + 6\right)\cdot 13^{2} + \left(9 a^{4} + 3 a^{3} + 11 a^{2} + 10 a + 6\right)\cdot 13^{3} + \left(5 a^{4} + 9 a^{3} + 12\right)\cdot 13^{4} + \left(8 a^{4} + 12 a^{3} + 9 a^{2} + 2 a + 5\right)\cdot 13^{5} + \left(9 a^{4} + 6 a^{3} + 5 a^{2} + 7 a + 10\right)\cdot 13^{6} + \left(12 a^{4} + 2 a^{3} + 3 a^{2} + 5 a + 8\right)\cdot 13^{7} + \left(6 a^{4} + 12 a^{3} + 8 a^{2} + a + 12\right)\cdot 13^{8} + \left(8 a^{3} + 5 a^{2} + 9 a + 7\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 7 + 13 + 3\cdot 13^{2} + 12\cdot 13^{3} + 10\cdot 13^{4} + 6\cdot 13^{5} + 9\cdot 13^{6} + 2\cdot 13^{7} + 7\cdot 13^{8} + 11\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 10 a^{4} + 12 a^{3} + a^{2} + 11 a + \left(10 a^{4} + 4 a^{3} + a + 11\right)\cdot 13 + \left(11 a^{4} + 5 a^{3} + 7 a^{2} + 2 a + 12\right)\cdot 13^{2} + \left(5 a^{4} + a^{3} + 8 a + 1\right)\cdot 13^{3} + \left(10 a^{4} + 4 a^{3} + 5 a^{2} + 4 a + 4\right)\cdot 13^{4} + \left(12 a^{4} + 4 a^{2} + 8 a + 4\right)\cdot 13^{5} + \left(12 a^{4} + 7 a^{3} + 2 a^{2} + 8 a\right)\cdot 13^{6} + \left(5 a^{4} + 10 a^{3} + 4 a^{2} + 8 a + 8\right)\cdot 13^{7} + \left(9 a^{4} + 3 a^{3} + 12 a^{2} + 7\right)\cdot 13^{8} + \left(3 a^{4} + 2 a^{3} + 12 a^{2} + a + 2\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 3 a^{4} + 6 a^{3} + 2 a^{2} + a + 12 + \left(12 a^{4} + 7 a^{3} + 12 a^{2} + 4 a + 12\right)\cdot 13 + \left(2 a^{4} + 3 a^{3} + 9 a^{2} + 12 a + 12\right)\cdot 13^{2} + \left(4 a^{4} + 6 a^{3} + 4 a^{2} + 2 a + 1\right)\cdot 13^{3} + \left(3 a^{4} + 7 a^{3} + 5 a^{2} + 11 a + 9\right)\cdot 13^{4} + \left(10 a^{4} + 2 a^{3} + 6 a^{2} + 7 a + 5\right)\cdot 13^{5} + \left(4 a^{4} + 5 a^{3} + a^{2} + 8 a + 10\right)\cdot 13^{6} + \left(2 a^{4} + 6 a^{3} + 2 a^{2} + 8 a + 7\right)\cdot 13^{7} + \left(2 a^{4} + 3 a^{3} + 3 a^{2} + 10 a + 12\right)\cdot 13^{8} + \left(11 a^{4} + 6 a^{3} + 10 a^{2} + 8\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 2 a^{4} + 9 a^{3} + a^{2} + 4 a + 1 + \left(11 a^{4} + 2 a^{3} + 9 a^{2} + 11 a + 12\right)\cdot 13 + \left(10 a^{4} + 8 a^{3} + 3 a^{2} + 5 a + 1\right)\cdot 13^{2} + \left(12 a^{4} + 2 a^{2} + 5 a + 6\right)\cdot 13^{3} + \left(8 a^{4} + 5 a^{3} + 10 a + 6\right)\cdot 13^{4} + \left(2 a^{4} + 5 a^{3} + 12 a^{2} + 2 a + 7\right)\cdot 13^{5} + \left(11 a^{4} + 5 a^{3} + 8 a^{2} + 5 a + 7\right)\cdot 13^{6} + \left(a^{4} + 11 a^{2} + 11 a + 3\right)\cdot 13^{7} + \left(6 a^{4} + 7 a^{3} + 10 a^{2} + a + 12\right)\cdot 13^{8} + \left(12 a^{4} + 4 a^{2} + a + 7\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 6 a^{4} + a^{3} + 10 a^{2} + 10 a + 6 + \left(2 a^{4} + 5 a^{3} + 5 a^{2} + 4 a + 2\right)\cdot 13 + \left(a^{4} + 10 a^{3} + 8 a^{2} + 10 a + 2\right)\cdot 13^{2} + \left(3 a^{4} + 3 a^{3} + 5 a^{2} + 2 a + 6\right)\cdot 13^{3} + \left(5 a^{4} + 12 a^{3} + 9 a^{2} + 12 a + 7\right)\cdot 13^{4} + \left(9 a^{4} + 12 a^{3} + 8 a^{2} + 12 a + 5\right)\cdot 13^{5} + \left(8 a^{4} + 3 a^{3} + a + 7\right)\cdot 13^{6} + \left(8 a^{4} + a^{3} + 2 a^{2} + 9 a + 4\right)\cdot 13^{7} + \left(3 a^{4} + 11 a^{3} + 2 a^{2} + 6 a + 4\right)\cdot 13^{8} + \left(9 a^{4} + 4 a^{3} + 7 a^{2} + 10 a\right)\cdot 13^{9} +O(13^{10})\) Copy content Toggle raw display

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

SizeOrderAction 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.