; F [7 l0 S W0 TQ<w> :=PolynomialRing(Q5);Q;+ Q$ F; K- x& q* T+ K
EquationOrder(Q5);5 f \, w" U& S3 i6 i6 }) S8 [
M:=MaximalOrder(Q5) ;. P; j$ q# ^$ z- M/ i
M; - K+ v6 g, ]* R$ F; y3 MNumberField(M);9 ?7 ^; v; K+ o" @
S1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888; ! P1 P9 {$ t: X: s! D" Q/ U3 ]IsQuadratic(Q5);. [1 n* r+ [2 \' {$ @, i
IsQuadratic(S1);$ c1 O/ I* p6 o
IsQuadratic(S4);- F) _6 e1 r7 q' \$ o. Q& k
IsQuadratic(S25); : [9 q; _/ o) H" R" kIsQuadratic(S625888888);7 l7 ?. f r% l4 k+ }+ r6 U) m4 u
Factorization(w^2+50); r3 B# u0 }3 p, r0 k
Discriminant(Q5) ;( _+ d! D7 Z* ~* {9 }; c
FundamentalUnit(Q5) ; / E7 w) c$ w/ q' v9 N$ WFundamentalUnit(M); ) y. M# s3 Q' {; EConductor(Q5) ; - k5 z" }8 v9 i" {1 Y& h# C" E( m4 Y& R/ A- X
Name(M, -50); 7 n! w5 ]: ^3 c9 ZConductor(M);" J. n( Z+ U9 D
ClassGroup(Q5) ; " |" r) J0 ^" J6 P- O& R) `
ClassGroup(M);$ m) C- W% Q' e6 a1 }3 @+ l
ClassNumber(Q5) ;% c* g: @7 P% }. j( e4 @
ClassNumber(M) ; # i8 y6 ]/ m1 Y4 q; NPicardGroup(M) ; # {( {6 {8 q5 v9 y8 GPicardNumber(M) ; P4 O) C* U' m2 I 3 I+ \: A c2 _: J5 U' {3 ?QuadraticClassGroupTwoPart(Q5); " Z4 W) B/ ]8 a) y' ? n( RQuadraticClassGroupTwoPart(M);1 ]/ C$ V) K, m: B: I
NormEquation(Q5, -50) ;8 U$ U& K! F/ m5 _
NormEquation(M, -50) ; 3 H" S- a; m5 B - n$ ~, S( O7 hQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field) U m7 k$ q# A+ w8 C/ X
Univariate Polynomial Ring in w over Q55 w, t5 Q- w+ n- p9 Q& k# L
Equation Order of conductor 1 in Q5 $ o5 l: \) {6 \' J6 K* IMaximal Equation Order of Q5 ! B3 c _0 F& ^) vQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field - D0 Q/ E1 ^2 k5 u/ LOrder of conductor 625888888 in Q56 K O. L% z+ F- J
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field % t" ]% V ^* d. F7 f9 M# R0 |true Maximal Equation Order of Q5 " M9 L0 Z3 d! M# b- B% ytrue Order of conductor 1 in Q5 ! F# j M5 E, t9 ltrue Order of conductor 1 in Q5 9 R3 w/ V3 R, P4 M8 l t( m. Ltrue Order of conductor 1 in Q5" ]* C! P5 }9 j7 K
[' N# U4 B/ b; S1 x) r8 B$ V7 a
<w - 5*Q5.1, 1>,. l' k2 l9 G7 k/ m4 C
<w + 5*Q5.1, 1> & I; Z/ X- {0 Q9 H, v+ c] 8 W# V- m& Q2 W4 b/ q- V-8 * [# k8 Q* ` T+ D& H9 K! E ^, B9 D- _+ S+ h) U& d>> FundamentalUnit(Q5) ; 1 g' k) u' S) |4 q3 f5 O ^- N k: M7 e7 H+ d
Runtime error in 'FundamentalUnit': Field must have positive discriminant5 e7 l5 O6 f; O# l; ^
2 F1 s9 c& z2 A' k# ~ T+ C# U/ A/ J, m
>> FundamentalUnit(M);3 M' F; J. k% \" v
^ " d& Y! M0 K1 jRuntime error in 'FundamentalUnit': Field must have positive discriminant& t( W+ A8 c4 H1 J
2 {$ U* t2 g1 s7 b4 ~+ N8 9 W8 ^% p1 E& {: w m, g6 k1 L, J2 y>> Name(M, -50); 4 c5 s; t/ m' E3 W0 a ^ ; f- U1 S- M. a' u) `6 Z- TRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1] * O' A6 u4 i2 r, e. A6 X1 N' n6 ]! B Z2 J# L9 m! p: f* m8 H
12 y. t, s4 a) r$ K. h
Abelian Group of order 1 / ~1 a( l4 J" d. H' j0 sMapping from: Abelian Group of order 1 to Set of ideals of M# Q4 v/ O; ^! f0 W+ k% t. C
Abelian Group of order 18 ~3 b7 R! F7 q* y5 V$ o: {
Mapping from: Abelian Group of order 1 to Set of ideals of M $ i F! _6 E% M1 i- z1* f9 Q. C& R( d' |
1 ! t, S* p3 M3 ~$ E& [" CAbelian Group of order 1; F) x' C$ p0 P$ j/ |
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no: e, p0 g) ~. ?4 ?
inverse] : l+ w. V, t$ _0 a% [2 m1 1 J0 L) O0 E1 c! XAbelian Group of order 1! G6 v# d, A0 r. y. [" a* }, d v
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant! |' Q5 u, X, ^6 W) j* N, ]
-8 given by a rule [no inverse]" E, s. W* M0 u7 Y0 l; o( b
Abelian Group of order 1 Q- L5 ?( T% TMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant " ^9 x+ Q/ G$ G' b2 X s) V0 Q% Y-8 given by a rule [no inverse] 6 U* f) Y' q. D7 [* X7 ]1 Bfalse; ?& P7 c" A4 j
false + u: M; E; l1 {) q3 ^1 v: T
看看-1.-3的两种：/ l6 w$ z6 B& a! o
8 q0 B. o$ }9 N/ }
Q5:=QuadraticField(-1) ;4 c" g2 a j& f3 q; L: e( j' p$ z
Q5; # W l, {0 Q6 N3 O, Q8 ~ 9 O. {8 t. s! Q* M" G8 n* BQ<w> :=PolynomialRing(Q5);Q; N; O: h+ P" \: }, K
EquationOrder(Q5);5 T+ x/ u/ e* _ o; s
M:=MaximalOrder(Q5) ; % A. g( L6 M B$ gM;* g& w" b; E/ Y0 z, Q6 f7 ~4 x# U
NumberField(M); 1 g& a. Q2 ]' HS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888; 9 E1 g8 M7 z' ]9 j9 ^6 v6 xIsQuadratic(Q5);; k Z* L# [9 E' G! B Q: _+ r7 p1 m
IsQuadratic(S1); ) N0 l# _2 O9 x, a5 AIsQuadratic(S4); 3 d& C$ b5 C8 K$ rIsQuadratic(S25); 6 M/ i& o- o2 I) dIsQuadratic(S625888888);6 ]) ~% s9 w0 x0 j8 }
Factorization(w^2+1); " z8 g. Z( n2 }
Discriminant(Q5) ; + k" m* t+ M: o! hFundamentalUnit(Q5) ; ; `+ o* Q* I* @FundamentalUnit(M);" n) V, f4 i' Q, |
Conductor(Q5) ;6 Q, o% L2 E8 s ^+ r
0 d# ~ \4 Y# T: O/ wName(M, -1); 4 R P3 Y$ q0 ~* A: M4 AConductor(M);$ M# l1 i) u. c9 R" y7 @2 h
ClassGroup(Q5) ; % }2 U7 t$ W1 c* dClassGroup(M);/ u6 U& `4 s/ `7 W
ClassNumber(Q5) ; 4 ?3 P4 M" Q2 a0 N. t3 z7 B: ?ClassNumber(M) ; : A/ k3 `2 V+ B/ W# ]& [' QPicardGroup(M) ;0 b% g9 G2 Z) u; l
PicardNumber(M) ;0 r7 T* U, F: b7 y) V+ l
5 B! k" \9 m; A RQuadraticClassGroupTwoPart(Q5);$ d# s& b8 V7 Y6 \ F. K! i7 f
QuadraticClassGroupTwoPart(M);: R( i0 x2 G4 d
NormEquation(Q5, -1) ; . n& _1 S8 a0 f/ `8 D4 W# `2 hNormEquation(M, -1) ; ; ? h: z, h! e. a# w9 J- H0 M. t4 W& Y# G- v
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field" J2 J C9 a) P2 P
Univariate Polynomial Ring in w over Q5 2 f+ i6 W& {( m3 X, ]( YEquation Order of conductor 1 in Q5 ; ?; W: a' c- q* E z m/ IMaximal Equation Order of Q5, A* u m) a7 w4 l
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field ) R" j- w' q' y( S7 qOrder of conductor 625888888 in Q50 k( ^3 X! M+ e/ c& d
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field# A/ O; P) E4 N; b. R
true Maximal Equation Order of Q5) T1 h1 A7 z& B! k+ X' B/ p" T6 g
true Order of conductor 1 in Q5' f; R( Y$ ?. o2 u. X3 z$ P$ w
true Order of conductor 1 in Q5 + y7 y, e2 T& E' M7 Ltrue Order of conductor 1 in Q5$ h. C I9 u, Y& L. p
[ 4 I) a7 B8 B& ^" x$ f& J <w - Q5.1, 1>,* Y# b% x+ ~, a$ t
<w + Q5.1, 1>4 @; f3 z5 A; A" l1 B+ V% R8 H- B( m
]! f) c% a4 x" g4 B! y
-4 ^+ j; z$ O. s- p $ b$ N! I7 O0 x>> FundamentalUnit(Q5) ; $ U7 l0 A0 G( N, @+ m ^7 J0 s8 B' j8 k$ [1 o) m
Runtime error in 'FundamentalUnit': Field must have positive discriminant - I2 c3 D* I% z: _) D6 E! H% H' d9 ]. y# X" p' U% Q% h) P# {
% r" b: Y* Q" l( p) y: Y>> FundamentalUnit(M);4 S' l+ k8 B4 s0 u4 E
^ & t5 @. v3 K! d/ b/ T3 q! iRuntime error in 'FundamentalUnit': Field must have positive discriminant 7 Z: r. k5 t/ `: p% Q ( m( F; p5 w2 w! s/ o) J; I46 U- l# x. N" i7 D% g) F
4 D Z- m0 q* g+ p/ q4 v
>> Name(M, -1);( C) }' i: z+ p
^ * K( ?6 J ] Q& dRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]5 x* s6 J% g; c
6 w! e6 X/ d: G, z# o
1! |7 D# N7 _$ f
Abelian Group of order 1 & z4 D* P( ^. w; f5 z& W4 f2 XMapping from: Abelian Group of order 1 to Set of ideals of M0 J) @" q# G& O
Abelian Group of order 1 ' a1 S) ^1 b: n/ RMapping from: Abelian Group of order 1 to Set of ideals of M # }8 G+ R( \* D& m1 # Y, L" W/ `2 @6 S9 Q5 X) _1 * V9 D$ K) H! vAbelian Group of order 15 q4 o8 J. O6 \9 o) m* l
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no; F, P& R8 \( d8 C: F/ J
inverse]6 }: t2 O' w) C0 x, K2 a
1. m( j2 c6 h0 Q( |$ s" `
Abelian Group of order 1. g6 X, `, ?8 y' U( t
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ( V7 i% Y/ `7 p ^-4 given by a rule [no inverse]) C4 k c, I/ w' v: d. B8 X4 |
Abelian Group of order 1 ' f- c- g" n. J7 I2 }+ m" _Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant7 y3 Q, m X0 a7 T
-4 given by a rule [no inverse]/ P% p- ~1 u z2 U
false' ]8 m$ d' M7 l- e3 T, t
false ( J1 u+ F6 B; d& z7 ?===============& [$ }% k9 U* V! }
% `- M' ?' K5 n6 ?Q5:=QuadraticField(-3) ;/ d" Y9 t- s. E( c/ o) R% _7 |
Q5;0 Z) A, R$ C$ D) N
0 C% J0 ^& ]3 q+ a @
Q<w> :=PolynomialRing(Q5);Q;, E6 g, i8 g& @$ Y8 w2 \# l2 w
EquationOrder(Q5);6 O, g' t' J% | R
M:=MaximalOrder(Q5) ;: }% Q0 Q$ X$ g
M; 3 }/ I& {" Q `6 \NumberField(M); , k$ `) U, i" u, rS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;/ L+ L, ]; G e( r
IsQuadratic(Q5);, m( s. G0 P) Q$ N6 v
IsQuadratic(S1); 9 N# G& l, o o4 f i1 ^IsQuadratic(S4);4 w9 e; }) F7 I2 k3 w" T9 b
IsQuadratic(S25); ! N* m/ U9 C# ^' G. z5 K( WIsQuadratic(S625888888); ' Y5 }7 o6 r% QFactorization(w^2+3); 9 p* Q ]& H; p1 a3 R
Discriminant(Q5) ;% ~( M5 O1 n+ A& K
FundamentalUnit(Q5) ;1 W5 u9 \% ^" ?# C) k+ U
FundamentalUnit(M);0 p4 {! D# H1 p$ \# U
Conductor(Q5) ;* F. ]: \# R2 R
9 |6 V' {* O0 cQuadraticClassGroupTwoPart(Q5);5 e2 `8 p* g I% l" Y' v
QuadraticClassGroupTwoPart(M);7 }, s- y @0 k! F5 q- I, i
NormEquation(Q5, -3) ; 1 Q3 Z* Z/ o. A# ONormEquation(M, -3) ; ' I3 d( n+ k, Z4 h+ H$ n# l4 P$ }& ~- |" |$ v! h9 W( x
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field9 i% y# s9 ^- j; M3 m
Univariate Polynomial Ring in w over Q5 1 T! V' Z6 ]7 L, f* p9 T! lEquation Order of conductor 2 in Q5 3 j( D5 {: O6 w( Y$ k9 V2 R j- ? CMaximal Order of Q5 & \+ ~) q4 }0 Y3 X! S/ N8 FQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field. j% K$ @3 v7 W" j% V0 t" B
Order of conductor 625888888 in Q51 t2 F1 ^: W4 I Q9 Q0 x
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field - K7 K. }2 p8 K) D8 w ], {/ ~) M6 utrue Maximal Order of Q5 % Z3 {2 M6 S- Btrue Order of conductor 16 in Q58 N$ E$ c# H7 x! t; N
true Order of conductor 625 in Q5" w" ~- ]; \9 c ?& m% m! U
true Order of conductor 391736900121876544 in Q58 D5 f2 q$ d+ c2 g. F- E
[2 h f- x4 j( T/ u+ y$ l
<w - Q5.1, 1>, 4 x0 g; f8 Y- d6 ~4 ^' w <w + Q5.1, 1> q( N I$ u" t( {) M: b! u
]$ J4 l* P. Z5 E+ }
-3 ; s2 i2 p0 K$ W+ V D' ?1 r & D1 \' b% g' ~" v& v- H# P>> FundamentalUnit(Q5) ;1 l. x- x& q( t% u& O
^ 8 T" F) S8 S3 a% j" g& w. ARuntime error in 'FundamentalUnit': Field must have positive discriminant # d. \) W) t" }/ x* }9 x; w" B/ [+ O9 A5 U
5 T* Q. ]3 T" n- Q9 ]>> FundamentalUnit(M);6 {( `! s: }' r/ x- t% U( x) H
^ / r' |0 U S0 V6 M% M5 H. P" g8 `8 k7 F& ?Runtime error in 'FundamentalUnit': Field must have positive discriminant3 G4 E% z" {/ }" K" c: n& P% @
$ l1 u. V# e r4 |" k
3 % |8 k: ?8 ?3 g( K: I5 ^2 e" v( m & G' q8 M* s5 X7 p. g1 _>> Name(M, -3);7 U1 T0 u* D& i5 ? F0 L% d' }
^# |' _) }* O0 n% w0 ]6 k9 I
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]8 D( P8 X+ }1 B8 t# U: b1 v9 S: X
; l P" z' H0 D: ~, U
18 m- K8 e+ J M; o, C2 z) |7 ~
Abelian Group of order 1 0 ?0 |% C# T" D$ a) V4 D6 D3 [Mapping from: Abelian Group of order 1 to Set of ideals of M ; V$ U! F- Z/ s- `( D- wAbelian Group of order 1: J2 L5 g$ R1 ] ^
Mapping from: Abelian Group of order 1 to Set of ideals of M 2 o( S8 d% `% Z1 F& b# H/ W% k: I1* g% H' Q' h1 {% Z) {6 r. f; K
1 ; E% X7 s C* IAbelian Group of order 1 6 f. J! \' M! f6 p/ v, u3 ]1 u6 w- rMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no: C4 S) `" F) ~" `& m/ X5 |
inverse] 4 k' L) E8 d; P1 # j) o9 @1 G4 ~- }2 a/ @Abelian Group of order 1 . r6 h" i) L+ d: }Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant4 _3 x- h- u* l3 R( | y6 s2 `- P
-3 given by a rule [no inverse] - R3 p# c& e6 {Abelian Group of order 1/ E9 D( w( m9 s* [
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant' J3 }6 P' I" P
-3 given by a rule [no inverse] 4 U F- q7 r! W3 U; J+ z+ z( ifalse4 ]: Q1 X- f' D% y0 }+ @
false
IP卡
狗仔卡
发表于 2012-1-4 17:41
转播
淘帖
分享
收藏
支持
反对
微信
提升卡
置顶卡
沉默卡
喧嚣卡
变色卡
抢沙发
显身卡
