本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 4 u. n! C- Q6 V. Q+ [% c
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Q5:=QuadraticField(5) ; $ D. B6 Y6 y; g/ oQ5;4 S4 l' g8 E4 l4 F6 ]
Q<w> :=PolynomialRing(Q5);Q; 0 l( P. Y" D0 S 7 }# {3 M3 O& {# C" m# e: p( sEquationOrder(Q5); y+ |' q( c6 s* v
M:=MaximalOrder(Q5) ; , k: v" R, l# l, D" i; w/ |; W9 Y9 pM; * ]2 f5 e: y) f* r# O9 r$ SNumberField(M); 8 j+ k, h5 `- h% bS1:=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;+ ]& e5 H2 H b6 D5 N0 u& g
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888); ) Q1 [+ t9 o' M7 MFactorization(w^2-3); + n" \; I. T+ N0 V, S+ qDiscriminant(Q5) ;0 N0 C) a& L; L1 ^5 e* m0 m
FundamentalUnit(Q5) ; ( \2 q" B* O, a }) ~FundamentalUnit(M);4 W y7 I: m6 r& Z1 {$ @
Conductor(Q5) ;" X2 |! F0 z# D3 S P& B
Name(Q5, 1);% U7 f7 W/ s: d# u2 V
Name(M, 1); ! h2 i5 I: }1 O- h: {8 Q0 XConductor(M);; E; J5 g9 k% G: M( t+ f
ClassGroup(Q5) ; ( ^7 K6 ~5 e) E" _5 ^7 {/ QClassGroup(M); & c1 [) x: g; }, A) O$ TClassNumber(Q5) ;& R9 _& Q; j/ e5 f
ClassNumber(M) ; 3 S/ U8 m2 N4 n+ y& T( b0 X( F5 }1 |" d5 j& F4 Y" F% R' s
PicardGroup(M) ; 3 j) ]7 q: z* CPicardNumber(M) ;" X/ n- L: i" \& p* T
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QuadraticClassGroupTwoPart(Q5); & ~) @, h/ J, f1 W6 v9 y4 RQuadraticClassGroupTwoPart(M);( u9 i0 C. H3 l; a* O
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NormEquation(Q5, 5) ;& G G; n/ a4 c8 H1 v. B, ]: n& l$ O. i D
NormEquation(M, 5) ; h- t! }$ X- i! b
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Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field 3 {( I& a1 H' V6 BUnivariate Polynomial Ring in w over Q5' s: Z1 s a- `) @- r7 w/ N
Equation Order of conductor 2 in Q5 1 o- N( l2 n: M; aMaximal Order of Q5 0 f2 Y' j" y) Z5 d1 {' w7 k, BQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field ' @& S! J% J+ i' M! A5 l1 x( FOrder of conductor 625888888 in Q56 O: j0 W8 b$ }
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field 8 y9 Y% K9 X% v" \( k e! ^& Y' jtrue Maximal Order of Q5) G6 ]' l9 ^7 D4 r. y& u, W
true Order of conductor 16 in Q5! b. E1 N1 B; b w8 d) f8 ?& `
true Order of conductor 625 in Q5$ c5 X. O; w1 H7 R
true Order of conductor 391736900121876544 in Q59 L- {' N$ h+ \ i! C% L
[ * C5 T) N" S8 ~1 w <w^2 - 3, 1>; N* n" `) J; e9 ~) K
] C. a. f, x" m6 s* I0 f/ q6 h5 , B$ g# Y6 ?3 T5 A: Q1/2*(-Q5.1 + 1) 5 g0 k% \4 r% W- ?-$.2 + 1 % c* |8 s3 R6 J$ N% ^5 + a1 H2 i% o5 m; bQ5.1 * h1 X/ s2 x! |$.2 8 g* C/ }' v$ h8 v1 3 M' A4 h+ G! @( x9 gAbelian Group of order 1% y! ]) t+ e R# u
Mapping from: Abelian Group of order 1 to Set of ideals of M; I( ?+ c5 x0 e1 k5 K# V4 ?/ e" N
Abelian Group of order 1( x) c' k2 k! X l- k
Mapping from: Abelian Group of order 1 to Set of ideals of M 8 e: P. H3 l* C+ r: E9 {! ^1/ g: Q2 e3 l0 h W+ }* J4 F
1; K5 P! J: I2 b2 U, i/ R; {
Abelian Group of order 1% o N1 Y( t3 _0 F- ^
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no! O0 X, w8 ^( F5 V' R
inverse] 7 y, I9 A: c- G1 L& I# c2 Z1 1 u S# {, w& W5 T4 ^6 ~0 c# uAbelian Group of order 1 . Y }" s$ p) K' W8 MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 8 I2 _9 r8 e, K0 m3 x5 given by a rule [no inverse] ; Q# g1 B0 M7 u5 CAbelian Group of order 1 ' I6 L- v' t1 d6 WMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant " o O( @! T4 I4 E' g5 given by a rule [no inverse]7 B) ?! p7 d! C8 c# B
true [ 1/2*(Q5.1 + 5) ]- g3 w2 K5 u5 ]) g4 \/ b( H1 ~- U
true [ -2*$.2 + 1 ] 7 }9 x7 s5 K6 J! r$ m% e; \; E% W p( K
9 [: _6 |, ^# l
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( u- K7 b9 h$ H. x: w3 }3 A9 A
, ?" [7 ]- U7 y0 f2 q0 D' l
( B8 f7 `- Y' K7 Y8 Y+ F6 Q 6 g( e' e, C& X0 Q6 ~1 K- ~( o" L7 k1 n) ?" j9 n
0 j, _* C' Y, m8 A
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4 }$ J7 h' J. [2 b/ G- x; J, F/ @============== ( [5 G5 T; z/ Y+ Q 5 Q3 N5 s. {- ?# b1 uQ5:=QuadraticField(50) ;; k: p! ~3 a( ]/ N" _/ f$ b+ T$ [
Q5; & f8 x& F5 A) d: l7 @: R* k 3 _7 i4 X: ^0 a! h" W& sQ<w> :=PolynomialRing(Q5);Q; & u9 h& U) A B% n f& v& J# MEquationOrder(Q5); ) |, Z6 a+ Z' j! g }; n+ M. E6 UM:=MaximalOrder(Q5) ; ( }9 u ~5 d, _/ W9 S qM; , Z; @. Z: n1 `' H: |5 sNumberField(M); ; w5 i4 q! k2 H- jS1:=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; R ^. S/ n% k: Z+ Y- L
IsQuadratic(Q5);+ R: h3 w' o! P6 u! U
IsQuadratic(S1);+ r! ]) O- p" h) e# m
IsQuadratic(S4);( d' r. P. g2 F
IsQuadratic(S25); % V, {# F7 _% O/ {0 x- R2 f2 ?IsQuadratic(S625888888);. q1 o' f5 t D1 h
Factorization(w^2-50); 3 k, f7 N/ r' X. H0 R
Discriminant(Q5) ; ! S M: w5 `5 B5 V3 @FundamentalUnit(Q5) ;: l) @$ D0 |. u" J, G" ?2 b' M
FundamentalUnit(M); , @' U' P4 B6 z* bConductor(Q5) ;: ` _, }5 r2 S* g7 B
' ^2 ~ f* C9 Z% NName(M, 50);7 J8 r- s, P4 ~
Conductor(M); 8 H4 ^4 V! L! u3 P( x7 ?! YClassGroup(Q5) ; 8 q6 M+ m7 M9 X. p2 yClassGroup(M);. \7 Y8 y5 q1 [" s5 }
ClassNumber(Q5) ; # X$ X4 T Q+ @/ j1 Z' JClassNumber(M) ;: M, s; |; S' C& G. D! v7 M% X$ A
PicardGroup(M) ;* l9 d3 v& Q$ N+ o9 q. B
PicardNumber(M) ;* V& }- p1 ]% E9 `% [, @
7 g; \' _) |3 G) j! h
QuadraticClassGroupTwoPart(Q5); 1 i3 c: O/ _, x2 M8 VQuadraticClassGroupTwoPart(M); $ O7 p% [0 e, h0 E! P" }1 KNormEquation(Q5, 50) ;) j% c E" ^# v6 Y
NormEquation(M, 50) ; * \+ K" h: Q& ] ! w# Y0 w) @: K7 `Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field a; \1 g6 K" X; \Univariate Polynomial Ring in w over Q5% j6 ~3 x* J+ d) y$ V1 L* i
Equation Order of conductor 1 in Q5 k! R+ g9 O7 M1 t9 w. HMaximal Equation Order of Q5 1 L( I2 A j. A. @4 JQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field ' U2 g0 U1 {6 `% a8 WOrder of conductor 625888888 in Q57 ?6 D/ i( y- e$ R% h: N- f% M0 n
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 9 P6 ]% R- n1 W4 j1 J# m$ rtrue Maximal Equation Order of Q59 l" B& ^# _# }3 X
true Order of conductor 1 in Q57 K4 x% p. ^9 V) k$ ^5 [
true Order of conductor 1 in Q5/ Q- r' U7 A% m: g8 M2 ` c% D
true Order of conductor 1 in Q52 F' h# M+ X4 O9 K% K
[5 |+ u8 i5 M6 O! a# M$ W
<w - 5*Q5.1, 1>, . ?0 l& d) L. X0 {# O' F <w + 5*Q5.1, 1>5 a7 p9 K! H( }! g, U0 J9 S
] ! a, T% g) f* X% E3 {1 R8 _: M: @: J9 b* i4 S, E; CQ5.1 + 1 1 _( d& `/ B. c9 H' Q' {( s2 Z$.2 + 1; j. i0 z; [% E8 e' W" p
8( Y' K, ^: |* y7 I: x
: X2 C" [0 Y1 D4 y: c, s>> Name(M, 50);/ E4 B2 L7 I: M) L; Y. I
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Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]* H* |; g& `0 @* o8 A! L
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Abelian Group of order 1# d l2 m' |+ g, N f, ]
Mapping from: Abelian Group of order 1 to Set of ideals of M D) i- v+ T6 `8 V7 R6 e- uAbelian Group of order 1 . f! h5 `; l6 r Z6 @; Y/ l5 eMapping from: Abelian Group of order 1 to Set of ideals of M 6 P3 r/ z5 c1 z* N& r+ L1. n- ~# a& ?' M. e3 u" G' W5 ^' Q
1 / [6 V0 Z1 |* e9 {Abelian Group of order 1( e1 K/ h# K) F& s
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no& `( N; T$ s" W
inverse] * a+ H" a& D" d3 x S: z18 o. G g3 ?" I2 |+ a& i
Abelian Group of order 1 5 ^8 ], T2 j# s( _3 k0 t4 V6 WMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ^ l, T/ y6 `- g. P5 Q
8 given by a rule [no inverse] * E6 W8 S7 Z2 ?- [$ _9 c7 j; {Abelian Group of order 1 . e/ F! h3 u1 i- YMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 1 U$ u* F* n \4 S0 H# N8 given by a rule [no inverse]# |& _, p& X( j5 m* |
true [ 5*Q5.1 + 10 ]- b2 {/ c3 `/ d0 P
true [ -5*$.2 ]