本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 8 [+ c0 z* c; w& C. F( Q3 p' Q
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Q5:=QuadraticField(5) ; . ~1 Q1 x) E9 {; L9 }Q5;* j. ]* g# g1 v! N1 P+ W
Q<w> :=PolynomialRing(Q5);Q;8 E4 ~" R N2 Y+ M# Y( \1 F
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EquationOrder(Q5); M& H2 T/ I @% R+ y3 h+ B tM:=MaximalOrder(Q5) ;2 O; a8 P0 D, m3 u3 l; \
M; + F' f7 |( l, {- P; n. \& mNumberField(M);4 @, t; L4 J# B$ a
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;! x. L, M* N( [ |) B& }7 J3 Z
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888); 1 |! U* `4 Z$ xFactorization(w^2-3); 0 G( D2 I; g, x1 a0 tDiscriminant(Q5) ; ' k2 r9 G8 u/ i, dFundamentalUnit(Q5) ; / p/ w/ T7 i* G! I# ^& G# z+ U) RFundamentalUnit(M); , h! F, ?5 k( g Q* RConductor(Q5) ; % x% I7 Z0 ^- c7 cName(Q5, 1);2 _0 M/ t- t+ H3 p
Name(M, 1);& d8 O2 t) S" B# z' i6 G
Conductor(M);; y3 w: Z0 Z5 Z( H
ClassGroup(Q5) ;( b% e/ b. A. t. z" y
ClassGroup(M);/ T7 k/ D2 w6 k( b; ~( C M# V. w
ClassNumber(Q5) ; & f6 i$ F, E1 f$ G% jClassNumber(M) ; ' Y, j" {# c4 q0 w6 c: r- f$ r: _; o' @+ `9 q" x3 y+ h; S
PicardGroup(M) ;) \; j2 k. L" I, X! t/ h8 B
PicardNumber(M) ;$ u3 d) r4 A* N! _; ^$ R6 w, Q
7 `9 T9 @& O* e% H. @6 F' I6 f* i. Q) d0 Z9 h
QuadraticClassGroupTwoPart(Q5);) r" Y3 d: b/ K+ ?0 F0 z* z4 E
QuadraticClassGroupTwoPart(M); / C6 W3 n; s1 E; O" O3 L7 B: c8 N! I: `3 Z% X+ L2 Z
0 v) a, t' b) t( V
NormEquation(Q5, 5) ; 0 v! N) o+ R) n! U3 }4 D3 w, HNormEquation(M, 5) ;- p4 P+ {% m) Z& x
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8 ]4 J& [6 W) aQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field 2 ^ i- i1 f0 iUnivariate Polynomial Ring in w over Q55 V3 ?5 ^. W, R K
Equation Order of conductor 2 in Q5 h; T4 c! ?/ J$ V% U3 A$ s# JMaximal Order of Q5 # C% @: [$ ^8 v4 FQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field # z) h, j+ |2 u. ]" KOrder of conductor 625888888 in Q51 ?% p- U3 J) a. j
true Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field : m7 i& Z P9 H) strue Maximal Order of Q57 ^, x7 k% ?2 l8 ^/ d
true Order of conductor 16 in Q5" b$ ]4 B9 A( Q0 r, G, A. K
true Order of conductor 625 in Q58 ?4 h: }4 n0 t V1 I3 v s
true Order of conductor 391736900121876544 in Q5& {( K2 `3 r( H
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<w^2 - 3, 1> 9 W! Q/ ]% O, \3 p( V4 E" e& F2 L$ C] & j# T1 j+ h1 K+ y, `5 ( K$ N0 b/ e M8 U/ p: G1/2*(-Q5.1 + 1)5 s0 o# f4 ^9 g: D1 ^
-$.2 + 14 Z! j% ^+ g8 [$ u# y. P; g! ]
5 3 G4 o+ {$ J* \ h$ ]- ~Q5.1: `% V' F5 T' {: U' o( \4 x3 g9 `
$.2% y$ Q- Z: Y" R( q. H4 p
1& I, P2 h& o. N1 n1 V2 y- y( }
Abelian Group of order 1# t* ?4 V: ]& t9 M4 `
Mapping from: Abelian Group of order 1 to Set of ideals of M" E2 c/ D7 u% a2 N6 @/ j
Abelian Group of order 1 " I6 u5 k @4 wMapping from: Abelian Group of order 1 to Set of ideals of M # I) ^& A( f$ ~8 s1/ x' R$ V0 k3 p, @% Y
1 6 ]# n) Q) s4 }# G8 E; |* {3 AAbelian Group of order 1' _+ n4 d0 l! K- R1 e3 x& }- ?: Z
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no7 @6 q# a1 {, z. t+ Y* e' L, ?$ h7 p
inverse], ]4 {- M+ \0 P# x" K0 j
1 % v6 n$ Y. k* @ w' j5 }Abelian Group of order 1" N n8 n% m! _/ S. Q+ h
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant* J a% [- S# Z* {5 c/ n
5 given by a rule [no inverse]9 Q2 }0 `& Y A& P& V0 g8 Z/ x' }; q
Abelian Group of order 1 $ o/ s; p- k/ j5 j1 n0 \; Z$ EMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant + O6 Z: I5 P: ] o2 q" \0 w1 ~0 K5 given by a rule [no inverse] ! b+ o& E" M$ P1 h. Dtrue [ 1/2*(Q5.1 + 5) ] * V3 S0 ~7 \+ V% wtrue [ -2*$.2 + 1 ]3 ~, H2 O+ c, G* l2 J1 t& H( G
" X; t' O" g% G; v- ~0 g8 f
1 N6 i5 S# p2 J& h9 K
) \2 o+ E3 u- J$ a: E! Y! @3 J! u4 T1 x, {+ M# K2 y2 {& w
|5 V5 S( l* j C& l+ o$ C' l
1 g* I/ G% s' u Y1 H
; o# \ o" ^, U/ M* J: h4 i
1 |3 k5 t) k' k, H9 V5 [, [
2 n, n+ e! J( r6 v6 h1 Z( U. z: N# v. L" ^9 U
% X( Y Y1 O; |8 S' I# Q6 }==============6 b; z% }& T+ |# `( O1 a- R
; N# A% W$ Z2 U$ x; X- ?0 S; L7 fQ5:=QuadraticField(50) ; / m$ u, o. ?! k8 s5 p4 j& ~" M2 SQ5; 6 R/ q7 D" [& D$ a: n. M8 S $ R5 M+ y3 f8 QQ<w> :=PolynomialRing(Q5);Q; * E8 R0 T6 S% U$ m8 {7 @EquationOrder(Q5);5 s6 V6 T, [1 z/ [0 \0 W# @
M:=MaximalOrder(Q5) ; # j! P- X0 d0 p6 y/ rM; 2 |' p9 M7 T# M5 J! fNumberField(M); & ?8 H/ F* L! k/ sS1:=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; z( F$ ^6 `) l9 u
IsQuadratic(Q5);- F( o6 M s3 }
IsQuadratic(S1);$ u3 I$ X$ a: B
IsQuadratic(S4); # B5 b' l, A. D9 f0 u$ HIsQuadratic(S25);, T$ |9 J# B/ D
IsQuadratic(S625888888);$ {7 W1 E, X/ m' c
Factorization(w^2-50); 9 J+ s! N! X, v- x9 `/ ]! A2 A
Discriminant(Q5) ; ! W6 G' o. a5 zFundamentalUnit(Q5) ;7 i6 H3 |- p" y* e+ I
FundamentalUnit(M);1 t+ K) H3 R; }8 T$ T
Conductor(Q5) ; + Y' E/ [+ T0 S! u3 [: C% H7 S/ i+ q* t/ s1 U
Name(M, 50); " u8 o |7 H* qConductor(M);. H& G! x4 }& j/ f
ClassGroup(Q5) ; . \9 g5 h! Z; }3 @' O- L- X( ^
ClassGroup(M);* M* j3 X, T/ d/ o
ClassNumber(Q5) ;* A( e. a/ I# f$ H" M [
ClassNumber(M) ; 1 r$ C9 ]( S- H. KPicardGroup(M) ; & C& W4 W* K' p' {3 G7 w$ y: vPicardNumber(M) ; + w1 }/ s; G) k3 b2 w: D0 C8 m" y; }4 b6 M/ n$ b5 Y
QuadraticClassGroupTwoPart(Q5); " E# F/ ?: Y$ K" S1 [0 p4 O4 a# V: @QuadraticClassGroupTwoPart(M);; u0 ]( b; Z* L4 {
NormEquation(Q5, 50) ; $ {' v4 q' q$ e' _8 O$ o9 eNormEquation(M, 50) ;# E' Q# v' d) Z) t
; ]' ^, A/ K% ~% I+ f+ Z
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field / r+ F# `9 {" H6 e0 R PUnivariate Polynomial Ring in w over Q5% ~( l$ l" U q3 v& W" f
Equation Order of conductor 1 in Q5" o$ ~5 E9 I {
Maximal Equation Order of Q5; l, p2 [5 V$ ~6 q% V( s" K
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 0 b. U# {7 ?1 C# n9 ?Order of conductor 625888888 in Q5, _, L4 \, `" | V& S" Y. L k' i/ @
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 9 s" T- y: s4 _+ O; ~3 ]3 ]true Maximal Equation Order of Q5 # ~6 z% n9 R/ N6 n' n. Ftrue Order of conductor 1 in Q5" M- M' }, v& l! M1 w) x
true Order of conductor 1 in Q5# G- p3 {! T6 R q5 G, S6 B
true Order of conductor 1 in Q5 1 P4 L3 Y+ |1 v U4 ^[8 y: S9 w$ j& F: h
<w - 5*Q5.1, 1>, T$ e9 d- u) i# Q
<w + 5*Q5.1, 1>1 J9 W5 \5 B& j$ S
] 5 V3 k8 n. _6 \5 U- {* [8 8 p! |, v5 p* E* WQ5.1 + 1 & F$ y7 x5 P1 e Y4 B# H$.2 + 1 . B5 @; Z5 V' A q8 4 ~7 t$ H# }' u) Z4 z; y' J. B l' \1 `5 A0 i9 m
>> Name(M, 50); 5 _! ~/ v3 i' l1 j5 C1 U$ ]6 v ^ ( h. v# g3 Y I0 @; g/ J7 ~Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]7 w2 D' x6 A0 Z8 L4 V
% X- q; t q \; W8 B; G9 K! T3 }1# W# l& n8 S/ U
Abelian Group of order 1 - e, \4 s' c/ i5 O6 W# w9 NMapping from: Abelian Group of order 1 to Set of ideals of M ) X: j- M/ \9 ~- mAbelian Group of order 1 + z. U! S% k, i! }" g* \Mapping from: Abelian Group of order 1 to Set of ideals of M 0 r5 M# c+ Z, S' x6 M2 }1 ! i- a6 {3 X6 Y% c9 K7 y5 O9 i12 a5 b, p9 f) b- z8 L+ x7 f' t# e
Abelian Group of order 1! }2 d" a0 H2 J
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no 2 O2 x1 K. e1 o' [! Cinverse]* d# j' {' O. a" `/ W1 v8 o
1 $ D8 ]( ^7 \9 k1 @/ rAbelian Group of order 1 2 q1 z) L+ l" @6 L! [& m* D" sMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant v" ?4 ~2 k6 [8 \6 j
8 given by a rule [no inverse] 2 Z6 V) f) _9 ?. Q' D( ?& X% }Abelian Group of order 1 ( m1 |, z: \9 D( `3 V$ ZMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant # c& n: w$ B/ Q8 given by a rule [no inverse]( M4 _3 ?, d4 V+ F
true [ 5*Q5.1 + 10 ] 0 _9 |1 I* T* x: X! X, F/ H5 mtrue [ -5*$.2 ]
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发表于 2012-1-4 14:05
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