Kernel: SageMath 8.1
Sage 8.1 on CoCalc
In [1]:
'SageMath version 8.1, Release Date: 2017-12-07'
In [2]:
'[ 1 0 -1]\n[ 0 1 2]\n[ 0 0 0]'
In [3]:
Eigenvalue: 8 with eigenvector: (1, 3, 4) of multiplicity 1
Eigenvalue: 2 with eigenvector: (1, 3, -2) of multiplicity 1
Eigenvalue: 0 with eigenvector: (1, -1, 0) of multiplicity 1
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x: (0.625, 1.4375, 1.25)
check A*x: (9.0, 5.0, -1.0)
bashslash: (0.625, 1.4375, 1.25)
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(9.0, 5.0, -1.0)
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(0.625, 1.4375, 1.25)
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libbraiding in sagemath: http://doc.sagemath.org/html/en/reference/groups/sage/groups/braid.html
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Braid group on 3 strands
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True
False
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True
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s1*s0
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False
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testing collaboration
2 * 1009
Elliptic Curves
In [13]:
In [14]:
In [48]:
In [49]:
224920054752209012888072239925666166152953315927462721091292220365989203840057376636804982037728842165211127754871180466185804122534771829601833896090835864196025436431924983737586223453054912366095785663715431040046594888989222421603285849332689122816156762538783671266032400455580674287929966023135712111798722104970689971267808493577063364619666993520659753007743124278585124546857412675047916113619801821701603298491596605616499083576796213009485555458658940236325255710647178563003216835067008947943476212542736824832348107381461218975410691680581263762269729207679626566396639963055239930575299610332004044486445038816865938212752679155455404917676116387237772041501081051928065361009924493262242457662069704890209189008254392022246914545263944336919637398245670306009262211730782937608082885675974083069527353881856749463300889196935129916598467151441999086699148318718203024413285031988755115636641407312419386479420956656440082347351337046535199009414178575816234342725422000488935597626813023611877232335981288141998222754683092883556669209212865773658833243333407531495199652510507949303438066553930402482227408453820470971508908699749310101805069233667208284943155721408078335784578449408
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Multivariate Polynomial Ring in x, y over Finite Field of size 13
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2*x + y
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12647
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35
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33
2/3
50
97
is b square? True
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1717162949
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'secret'
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2018
In [26]:
1450942
In [27]:
2018
2 * 1009
15511210043330985984000000
[22, 10, 6, 3, 2, 1, 1, 1]
2011
2003
[1, 2, 4, 7, 14, 28] 56 56
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d: 2, u: 1, v: -1
True
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m 1: 1
m 2: 499
m 3: 665
m 4: 748
m 5: 399
m 6: 831
m 7: 285
m 8: 374
m 9: 554
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[2, 3, 79, 42557]
In [31]:
6638736
In [32]:
6724006
In [33]:
2
1
[1, 13, 78, 286, 715, 1287, 1716, 1716, 1287, 715, 286, 78, 13, 1]
[1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1]
[1, -1, 1, 1, -1, -1, -1, -1, 1, 1, -1, 1]
-23
[[4], [3, 1], [2, 2], [2, 1, 1], [1, 1, 1, 1]]
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11-adic Field with capped relative precision 20
4 + 4*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9 + 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16 + 3*11^17 + 11^18 + 7*11^19 + O(11^20)
10*11^-2 + 5*11^-1 + 4 + 2*11 + O(11^18)
10*11^-2 + 5*11^-1 + 8 + 6*11 + 11^2 + 7*11^3 + 9*11^5 + 5*11^6 + 4*11^7 + 8*11^8 + 7*11^9 + 9*11^10 + 3*11^11 + 10*11^12 + 11^13 + 5*11^14 + 6*11^15 + 2*11^16 + 3*11^17 + O(11^18)
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Galois group PARI group [6, -1, 2, "S3"] of degree 3 of the Number Field in a with defining polynomial x^3 + x^2 - 2*x + 8
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[1,
5,
9,
13,
17,
21,
25,
29,
33,
37,
41,
45,
49,
53,
57,
61,
65,
69,
73,
77,
81,
85,
89,
93,
97]
formula
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[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
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[1 0 1]
[0 1 0]
[0 0 0]
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In [40]:
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
[1 0 1]
[0 1 0]
[1 0 1]
[1 0 1]
[0 1 0]
[0 0 0]
In [41]:
8
In [42]:
[ 2]
[ 6]
[10]
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[ 6 8 10 12]
In [44]:
[70]
In [45]:
[ 5]
[13]
[21]
In [46]:
[ 1 4 9 16]
[ 25 36 49 64]
[ 81 100 121 144]
In [59]:
[2 3]
[6 7]
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