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楼主 |
发表于 2011-7-4 15:44
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群常识常用计算方法：
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3
阿群初等因子不变因子：
s1:=Sym({ 0..8 });
A0:= AbelianGroup([2^7,3^4,5,7]);
A0;
s2:=Sym({ 0..12 });
s2;
A1:= AbelianGroup([2^10,3^5,5^2,7^11,13]);
A1;
A2:= AbelianGroup([30,140,250]);
A2;
Symmetric group s1 acting on a set of cardinality 9
Order = 362880 = 2^7 * 3^4 * 5 * 7
Abelian Group isomorphic to Z/362880
Defined on 4 generators
Relations:
128*A0.1 = 0
81*A0.2 = 0
5*A0.3 = 0
7*A0.4 = 0
Symmetric group s2 acting on a set of cardinality 13
Order = 2^10 * 3^5 * 5^2 * 7 * 11 * 13
Abelian Group isomorphic to Z/159907204637107200
Defined on 5 generators
Relations:
1024*A1.1 = 0
243*A1.2 = 0
25*A1.3 = 0
1977326743*A1.4 = 0
13*A1.5 = 0
Abelian Group isomorphic to Z/10 + Z/10 + Z/10500
Defined on 3 generators
Relations:
30*A2.1 = 0
140*A2.2 = 0
250*A2.3 = 0
A2:= AbelianGroup([30,140,250]);
A2;
FactoredOrder(A2);
Abelian Group isomorphic to Z/10 + Z/10 + Z/10500
Defined on 3 generators
Relations:
30*A2.1 = 0
140*A2.2 = 0
250*A2.3 = 0
[ <2, 4>, <3, 1>, <5, 5>, <7, 1> ]
阶为1050000的群有：初等因子：
1:Z2+Z2+Z2+Z2+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7 初等群
2:Z4+Z2+Z2+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7
3:Z8+Z2+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7
4:Z16+ Z3 +Z5+Z5+Z5+Z5+Z5 +Z7
5:Z2+Z2+Z2+Z2+ Z3 +Z25+Z5+Z5+Z5 +Z7
6:Z2+Z2+Z2+Z2+ Z3 +Z75+Z5+Z5 +Z7
7:Z2+Z2+Z2+Z2+ Z3 +Z375+Z5 +Z7
8:Z2+Z2+Z2+Z2+ Z3 +Z375*5 +Z7
9:Z4+Z2+Z2+ Z3 +Z25+Z5+Z5+Z5 +Z7
.............
共有4*5=20种
1:转不变因子：
12222 11113 55555 11117=Z5+Z/10+Z/10+Z/10+Z210
9：转不变因子:
1 2 2 4 1 1 1 3 5 5 5 25 1 1 1 7=Z5+Z/10+Z/10+Z/2100
验：对！同构可关系不同，1和9中一个是11/5生成元、一个11/4生成元
A22:= AbelianGroup([2,2,2,2,3,5,5,5,5,5,7]);
A22;
A22:= AbelianGroup([4,2,2,2,3,25,5,5,5,7]);
A22;
A2:= AbelianGroup([5,10,10,10,210]);
A2;
A22:= AbelianGroup([5,10,10,10,210]);
A22;
Abelian Group isomorphic to Z/5 + Z/10 + Z/10 + Z/10 + Z/210
Defined on 11 generators
Relations:
2*A22.1 = 0
2*A22.2 = 0
2*A22.3 = 0
2*A22.4 = 0
3*A22.5 = 0
5*A22.6 = 0
5*A22.7 = 0
5*A22.8 = 0
5*A22.9 = 0
5*A22.10 = 0
7*A22.11 = 0
Abelian Group isomorphic to Z/10 + Z/10 + Z/10 + Z/2100
Defined on 10 generators
Relations:
4*A22.1 = 0
2*A22.2 = 0
2*A22.3 = 0
2*A22.4 = 0
3*A22.5 = 0
25*A22.6 = 0
5*A22.7 = 0
5*A22.8 = 0
5*A22.9 = 0
7*A22.10 = 0
Abelian Group isomorphic to Z/5 + Z/10 + Z/10 + Z/10 + Z/210
Defined on 5 generators
Relations:
5*A2.1 = 0
10*A2.2 = 0
10*A2.3 = 0
10*A2.4 = 0
210*A2.5 = 0
4
有限自由阿群及ECC中的RANK理解：
就是有整数无限群Z的，有一个Z，RANK=1，n个，RANK=n,
有限生成阿群就是有整数群Zm间直和，叫挠子群----都是阶有限的，在ECC里没MOD P前就是画直线只能求有限个新点，
有限生成阿群也有可能加几个无限群Z的，就叫有限生成自由阿群，象ECC里的曲线现在发现最高加了28个无限群Z，在ECC里就是画直线能求无限个新点,这28个点（生成元）如上
不过ECC曲线MOD P后就没无限群Z这直和项了
有限生成阿群麻烦在初等因子---就是Zm各项的m求法
也可表为不变因子之直和
初等因子就是素数次幂（》=1）分解之和----------不变因子转换麻烦点等看懂了贴这
RANK=3
E3:= EllipticCurve([0,0,0,-82,0]);
E3;
PointsAtInfinity(E3);
TorsionSubgroup(E3);
NumberOfGenerators(E3);
Generators(E3) ;
Rank(E3);
MordellWeilShaInformation(E3);
AbelianGroup(E3);
P1:=E3![-8,-12];
Order(P1);
P2:=E3![-1,-9];
Order(P2);
P3:=E3![-9,-3];
Order(P3);
P4:=E3![0,0];
Order(P4);
P5:=E3![49/4,231/8];
Order(P5);
P6:=E3![41/4,123/8];
Order(P6);
Elliptic Curve defined by y^2 = x^3 - 82*x over Rational Field
{@ (0 : 1 : 0) @}
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
2*$.1 = 0
4
[ (0 : 0 : 1), (-8 : 12 : 1), (-1 : -9 : 1), (-9 : -3 : 1) ]
3
Torsion Subgroup = Z/2
Analytic rank = 3
The 2-Selmer group has rank 4
Found a point of infinite order.
Found 2 independent points.
Found 3 independent points.
After 2-descent:
3 <= Rank(E) <= 3
Sha(E)[2] is trivial
(Searched up to height 100 on the 2-coverings.)
[ 3, 3 ]
[ (0 : 0 : 1), (49/4 : 231/8 : 1), (41/4 : 123/8 : 1), (-9 : 3 : 1) ]
[
<2, [ 0, 0 ]>
]
Abelian Group isomorphic to Z/2 + Z + Z + ZDefined on 4 generators
Relations:
2*$.1 = 0
0 0就是无限远点，3个Z群，生成元P1/P2/P3
0
0
2 2阶点两个挠子群,生成元P4
0 还给了两能生成 Z群的点-----分数也行，只要是有理点
0
-----------
100000*P4;挠子群Z转不出去
100001*P4;
(0 : 1 : 0)
(0 : 0 : 1)
Z群就不同了，10倍就天文数字了，可至无穷啊
(5329/144 : 377191/1728 : 1)
(-118764872/42003361 : -3937849795044/272223782641 : 1)
(905925300579649/81949277077056 : 15640109412983368378849/741852714479323912704
: 1)
(-2343465084196597805000/58051145063946951161569 :
-25447354793050005176981463556943100/13986729827640029730757694665036753 : 1)
(6302474098508073788197910531446609/651331880237428048545105511066896 :
-176342732523930673371934739457017411658357216614489/16622774127029891762931364\
252673114873466120250944 : 1)
(-3813430085070199683943976557798873904463613832/228336826875138911896596598141\
8615777603848641 : 125494893852863155426100974766561536605189928217522571232033\
4800833324/10910984586392115360355313355505782235146810851274627664315031114256\
1 : 1)
(1880701588819492767318652030277859714069703191156088494052609/8018344141367449\
6508774280969881993157804843783416953223424 :
-237919946592218328913913464119432611871936626120863837945348807020008535118330\
0600626847871/22705289196485300714253305436925878920371641314520424153731523057\
840376182235816117587968 : 1)
(-18815230184579826991959942950507242936389088171715617440267931518219770954888\
/2785944412729486384827083679528629259002420636194629323893461343681407696961 :
2305204200717457550192851363078188878716253530093747095851230611288478913625408\
572457649696650868753188600387657492/147047847011317425399566847867848018869257\
649673254707518241304784767107976145675764728694749353143581342134675041 : 1)
(190909762816982266589447623951781585733423203877205763763861577911709314049308\
99671515742264401/3759205612578986117009614770431984774241158894350139721932557\
4936713501758087453032010090000 : -26373827153651235352076330133554362293020016\
8338935545161185268632853079379474560027219993229788750399658301115811130372933\
8142837271804656601/23048577162798624548776580990172190660088839559951272602553\
3782892120933828081749097515442095457075887927302595340997837489878439223000000
: 1)
1000*P1;过5万位了
The output is too long and has been truncated.
RANK=4
E3:= EllipticCurve([0,-1,0,- 24649,1355209]);
E3;
TorsionSubgroup(E3);
NumberOfGenerators(E3);
Generators(E3) ;
NumberOfGenerators(E3);
Rank(E3);
AbelianGroup(E3);
Elliptic Curve defined by y^2 = x^3 - x^2 - 24649*x + 1355209 over Rational
Field
Abelian Group isomorphic to Z/2 + Z/2
Defined on 2 generators
Relations:
2*$.1 = 0
2*$.2 = 0
6
[ (67 : 0 : 1), (113 : 0 : 1), (149 : -984 : 1), (-15 : 1312 : 1), (313 : 4920 :
1), (-56 : -1599 : 1) ]
6
4
Abelian Group isomorphic to Z/2 + Z/2 + Z + Z + Z + ZDefined on 6 generators
Relations:
2*$.1 = 0
2*$.2 = 0
E3:= EllipticCurve([0,-1,0,- 24649,1355209]);
E3;
Generators(E3) ;
NumberOfGenerators(E3);
Rank(E3);
AbelianGroup(E3);
Order(E3);
P1:=E3![67,0];
Order(P1);
P2:=E3![113,0];
Order(P2);
P3:=E3![149,-984];
Order(P3);
P4:=E3![-15,1312];
Order(P4);
P5:=E3![313,4920];
Order(P5);
P6:=E3![-56,-1599];
Order(P6);
Elliptic Curve defined by y^2 = x^3 - x^2 - 24649*x + 1355209 over Rational
Field
[ (67 : 0 : 1), (113 : 0 : 1), (149 : -984 : 1), (-15 : 1312 : 1), (313 : 4920 :
1), (-56 : -1599 : 1) ]
6
4
Abelian Group isomorphic to Z/2 + Z/2 + Z + Z + Z + Z
Defined on 6 generators
Relations:
2*$.1 = 0
2*$.2 = 0
>> Order(E3);
^
Runtime error in ';Order';: Algorithm does not work for this ring
2------2阶点两个挠子群,生成元P1/P2
2
0
0
0
0-----------0就是无限远点，四个Z群，生成元P3/P4/P5/P6
[br][br]-=-=-=-=- 以下内容由 cjsh 在 时添加 -=-=-=-=-
一般线性群或者它的一些子群是非交换的，只有有限生成交换群采有秩。非交换的群是没有的。 |
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发表于 2011-7-4 15:39
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