本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 0 Q! W9 ~" c( E2 n" j/ f l9 [) J& n' |6 ~2 ]% r0 `& T( A
Q5:=QuadraticField(-5) ; % s& U+ ^, W8 M) a; ZQ5;/ x' l$ x9 B. I4 A8 T+ h
0 @, F+ P+ n! ?
Q<w> :=PolynomialRing(Q5);Q;) d6 p P6 B6 v4 W
EquationOrder(Q5);2 a8 {( u% R2 h+ l
M:=MaximalOrder(Q5) ; 1 C0 |: @' c' H' UM; 8 i* v, T/ W2 r, |0 cNumberField(M); E4 V0 g6 p; QS1:=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; ; c: E8 S, M3 D2 R9 BIsQuadratic(Q5); * {- D0 \9 X; [4 pIsQuadratic(S1); . M6 L+ }2 U/ N8 x x* QIsQuadratic(S4);( h. \( H- T3 t3 m
IsQuadratic(S25);. l: V. B$ i! k2 u. g' R
IsQuadratic(S625888888); ' o# F" _, f% |Factorization(w^2+5); + T! X' L# Y# g$ n9 Q
Discriminant(Q5) ;7 W. ]8 C* Q- \) C- B
FundamentalUnit(Q5) ;" t/ Q6 p4 i! t: U+ f
FundamentalUnit(M);2 ?5 y3 } V4 ?9 P3 v* n/ }! x
Conductor(Q5) ; , L: U u4 N# _& Z1 L# C5 H& b " [+ x4 i$ |% z1 \, Y9 W6 `% X. HName(M, -5); : @! E. P* F+ w1 v) DConductor(M);* v- e' I8 i7 B( o2 z
ClassGroup(Q5) ; % v5 t' ^8 \, R: V% L, w X8 ZClassGroup(M);# g% Q5 s. |7 z1 F1 [ A$ K' g
ClassNumber(Q5) ;; \* j- N, o% ]6 ^
ClassNumber(M) ; 8 j3 S+ O/ z/ ~# sPicardGroup(M) ;, G4 d% ^# e/ |. ~6 @
PicardNumber(M) ; % ~- r# W" T& U5 U6 z2 I % W6 Q- n* h* j2 dQuadraticClassGroupTwoPart(Q5); 9 L. b/ w0 M2 G/ hQuadraticClassGroupTwoPart(M);7 M; w* R% I9 }; Q, @! b" y
NormEquation(Q5, -5) ; 2 ~, Z0 F) F5 U3 l* \) R) _8 [NormEquation(M, -5) ; # s1 `( g; p. x; [ K/ m# AQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field 5 H& ?) }& X }% I: FUnivariate Polynomial Ring in w over Q5, e6 A% b( U) V& U4 B* X$ Q
Equation Order of conductor 1 in Q5 * V9 f! T. G* D8 K5 B/ n. @Maximal Equation Order of Q5 7 J/ @4 U% N/ X) [4 ^Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field 3 i9 t6 U( t O- @5 \Order of conductor 625888888 in Q51 E2 c4 ]; G3 G
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field! @+ n3 }8 O ?# E/ Z0 H
true Maximal Equation Order of Q5 ( X) k* |; H( j# m, Z9 ~true Order of conductor 1 in Q5& b' o, _' p$ w8 t* G4 w3 Y- }5 `
true Order of conductor 1 in Q5 # g7 R! B/ F) X9 a9 b2 ^+ Ztrue Order of conductor 1 in Q5 - z" Q5 e9 k1 S& P[% G; J& D) U# i
<w - Q5.1, 1>, 8 x5 c. a' d4 T7 w/ T <w + Q5.1, 1>0 B" f8 `2 z0 a3 \/ f6 e# p) M
] ! L6 F# v9 ]* V1 t# A7 o-20 & h1 X' C. K6 z4 o ) t3 k4 A" c9 A; A q3 L>> FundamentalUnit(Q5) ;4 H0 p/ y7 s/ w, g
^ : p+ `+ j' c' R" `9 ZRuntime error in 'FundamentalUnit': Field must have positive discriminant , |: A6 s! P% r0 d: m# R; x, k; A; V8 Z1 ^) q
9 H( w' `% v2 s
>> FundamentalUnit(M); 8 E, z; C# t' K ^ 6 r! v/ U9 k- q8 C* ^ U& wRuntime error in 'FundamentalUnit': Field must have positive discriminant + C- S: Y8 o7 Z0 Y) Y, U5 _: y. e6 O- t# X
20 ; I+ |4 ]& P. o" i: A$ T' w & ]: l4 A( c$ `; I. P9 B>> Name(M, -5);) ^/ m' ^% }( h! ]- q6 s
^ ) F: @& e, ^4 S: {: B' xRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1] 6 Z5 Y9 Q$ [) i1 X7 s" y3 ?1 }4 E1 h
1 & w. I% W, k5 {6 {Abelian Group isomorphic to Z/2 0 B% y7 i1 }* I+ X! s4 C: Q! E7 TDefined on 1 generator + ~ E8 k5 e) D u& n+ M9 ~, xRelations:2 i% e3 q; s P) O I" K
2*$.1 = 09 P; l9 W% a. R+ n4 Y9 y7 ?
Mapping from: Abelian Group isomorphic to Z/2 ; E* `' q Q/ v; Q5 @Defined on 1 generator ; ^2 I. B( p6 w' Z+ |Relations: e& }8 j; H4 T- v1 Q+ s% f 2*$.1 = 0 to Set of ideals of M: n* B% p" e8 ~/ l( G1 s+ ?
Abelian Group isomorphic to Z/2 d o! a( A8 k% s, J
Defined on 1 generator5 G- Z7 w2 d" Q) U: M1 k( w) a1 G
Relations:" [9 G6 }7 q: ^# }) K# t
2*$.1 = 0 5 H# W$ }; T X" J: \Mapping from: Abelian Group isomorphic to Z/2 % i( O6 [% g/ }Defined on 1 generator+ V/ A8 G5 E/ p* [* ?% ^
Relations:# ]/ r9 u9 c k+ X
2*$.1 = 0 to Set of ideals of M9 T" J: t# q# T' R* E( g
2# {8 r1 J R- u% {
28 t5 ?$ u. |- L
Abelian Group isomorphic to Z/2. }; ]5 C0 l1 a1 `4 b
Defined on 1 generator - L/ p9 H ]5 P3 ^& x9 jRelations:# E* I! H+ U" F. @. u
2*$.1 = 0! D8 d$ p2 ]7 ~% R; P3 L
Mapping from: Abelian Group isomorphic to Z/2' M' U; Z* }5 m T0 m
Defined on 1 generator 1 v6 g8 E4 l1 y$ {Relations: + m7 l# C, g# V: o 2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]2 G- F7 M9 i0 f5 G3 }
2 4 G; L# h1 T$ q9 a9 eAbelian Group isomorphic to Z/2 ! x: d( o( _$ @4 A4 aDefined on 1 generator ; e! F! D* g; l, l! @Relations:: M0 |/ C+ V; j. l* R$ J
2*$.1 = 0 $ Q% O) L t# n- S5 JMapping from: Abelian Group isomorphic to Z/28 j7 W. p. |7 F' X
Defined on 1 generator$ t2 C* ^0 f; L
Relations:# ^ ?, V0 X2 j/ Q' f! f
2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no + P X& C. J) Z" j! k( M- K' T
inverse]/ A! v5 ~7 {% A, g
Abelian Group isomorphic to Z/2 , I. u5 O( m7 Q# C3 K% i9 Q& z4 ?Defined on 1 generator * X! h8 d1 E$ @/ B( {; cRelations: , y) R+ t8 L7 J+ D4 Z+ D 2*$.1 = 0 1 P$ h* `0 l! U9 n& VMapping from: Abelian Group isomorphic to Z/2 " Y* k2 r0 M, l* `& ], ^3 W/ jDefined on 1 generator: A* z/ W) e+ M; M4 A# ~" M
Relations: , k% [) M- Q0 i- a 2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no 7 c; z3 e1 W: }+ T
inverse]3 f2 b8 s+ ?& L6 \( N
false ( k: ^) l* M- f) sfalse# X% D% f) e* {/ H: E
============== # _: B) m0 A4 ^1 t8 D9 e" E5 B Z" V" I: L& N6 O4 A
$ [1 ], e& L# AQ5:=QuadraticField(-50) ;- G* a9 a9 E6 s6 h ~" f Y* |
Q5;5 P( ?7 ^5 F+ m5 l6 L
4 d6 ], N6 v% l2 GQ<w> :=PolynomialRing(Q5);Q;, }1 O3 V# J# i" y& @
EquationOrder(Q5);: y5 B$ S/ f& ^$ Q
M:=MaximalOrder(Q5) ; . F# Z6 a% r: _2 Q. M3 ]M; : b5 a f- F2 `" C6 u8 R6 ^- QNumberField(M);# ?7 U( Y# S/ k3 x+ W
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;' t4 e4 ^, w9 s: D* A% H' |
IsQuadratic(Q5); ! Z- ?- G+ b+ A: n0 L- W# ?IsQuadratic(S1);% H# x+ V' n0 `" c: w; R T: e
IsQuadratic(S4);6 f( w9 h. b* c4 E6 \2 C
IsQuadratic(S25);3 v) H/ h- k' j& Y+ H
IsQuadratic(S625888888);4 C- @: a6 s% ]- ]" S' O1 p
Factorization(w^2+50); - H. R: M2 X# `/ T& W
Discriminant(Q5) ;( n3 f2 Z$ g% B; ^- B- k
FundamentalUnit(Q5) ;2 T k/ N' \* d5 ?
FundamentalUnit(M); 4 O: d% q, P& tConductor(Q5) ;$ F1 B3 |5 N$ R6 k) W# R
% }' y! F' ]# y2 sName(M, -50);& d7 {) |6 |: O0 R8 M% A
Conductor(M); ! t8 O5 `9 H6 [- ?( Y6 C3 \3 wClassGroup(Q5) ; , }* G* s; V1 f
ClassGroup(M);/ Y6 H5 z8 u6 J7 L" V* G Q
ClassNumber(Q5) ; # n$ ^, V7 o7 D0 y, XClassNumber(M) ;+ {; G4 {8 Y' Y: f% c/ I
PicardGroup(M) ; 5 O# P* _& p/ F& t4 rPicardNumber(M) ; ' J7 p* A0 M. g' Y& o5 N8 o- t8 x$ H; \' h
QuadraticClassGroupTwoPart(Q5); 9 `. n/ A2 A: h; x0 L) ZQuadraticClassGroupTwoPart(M); 1 j; p3 [; A9 Y; A- @NormEquation(Q5, -50) ;2 |: z/ f: p% v6 _# {7 E$ A# r& E
NormEquation(M, -50) ; 1 B6 y+ P% M0 m+ e, f5 U w1 E; {8 D; g+ O
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field 8 b, H7 g& Q7 f# k/ _7 fUnivariate Polynomial Ring in w over Q58 h7 V a, c( n& ]2 x3 d
Equation Order of conductor 1 in Q5# N& A6 i0 X9 a
Maximal Equation Order of Q5/ ] `" K h8 J' p* C2 a j0 h
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field 4 U% y& n3 b) D6 l' \' ^/ lOrder of conductor 625888888 in Q5 : K, |4 V# f0 h' P3 Htrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field5 v" Y+ X* Z A: G4 q
true Maximal Equation Order of Q5+ v) A% I: }9 N% H6 e7 `
true Order of conductor 1 in Q5- E" R& _8 s) |* c% S; k3 ]
true Order of conductor 1 in Q5, k4 f3 `8 R; }& v9 D1 R' E
true Order of conductor 1 in Q5 $ N& W% [" j7 d$ |( ~. c[. B8 W5 ^+ Q7 G' H1 F
<w - 5*Q5.1, 1>, 7 c, a' k& Q" m9 u$ f1 p1 x <w + 5*Q5.1, 1># R& o4 h3 y H+ `% f4 d
]$ ~; \, E1 {/ H& {$ w8 F9 o
-8 P3 `; Y+ f! r3 } x Z7 A7 k! s3 k, J) _) v
>> FundamentalUnit(Q5) ; 0 C. y; ]! y' X' [$ `% g ^ / H/ ?+ a+ L) h/ K4 [% t# JRuntime error in 'FundamentalUnit': Field must have positive discriminant $ n! B B+ T6 `2 v/ D Y9 M# x& d* \2 ~3 B% T0 l
f- ?9 f/ |# g
>> FundamentalUnit(M);& X* C. k% j/ G" ^
^3 |0 j, j" `, }9 z- T$ h
Runtime error in 'FundamentalUnit': Field must have positive discriminant 5 Z- s/ P0 G- ?9 l. A* w1 v% H, K& \, V, v+ S4 `
8- d; h0 A5 c6 x+ m
$ I) J A& V7 h>> Name(M, -50);; U9 w c" A) t( i" i/ _9 e& t G
^3 |% o- C+ V6 o( s; r1 g
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1] / @3 ]" W. \$ }9 z6 g7 h. L0 q) Q2 Y4 ^2 ?7 B
1 7 U: S1 r4 N) ~# W* OAbelian Group of order 1& C' V4 N2 r8 C
Mapping from: Abelian Group of order 1 to Set of ideals of M+ a" S% K7 c2 v" V' ~8 [, I8 f% B
Abelian Group of order 1' v; u& S/ L7 I" ~& |& A: W/ f
Mapping from: Abelian Group of order 1 to Set of ideals of M 9 _2 O) R+ X) R: T2 Q# d6 C1 8 t' e8 U# [; t14 u% m$ M9 }& H" h7 F6 C
Abelian Group of order 1% D) H k# u1 c* P# n! R9 R9 \
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no& e. H( {, x& Y# P, v8 a: H
inverse] 4 S" G' B; J7 \18 g6 P, p) ]1 A/ {; T, n
Abelian Group of order 1 # c7 o8 l6 w5 |9 p. b" _1 KMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ' m0 g4 {, N4 _# ]5 l5 Z7 y-8 given by a rule [no inverse]1 `2 S. [8 R0 [7 \3 T7 Y
Abelian Group of order 1' c; r/ h3 i* u0 c: l
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ) B' L" m& V' f' p-8 given by a rule [no inverse] Z' }) X( i1 C! ]5 N0 p) d
false 9 v7 |* t, N5 c& F V1 Sfalse! n1 [4 k" z) m- _
看看-1.-3的两种： 0 _; G& K% ^, X; J: m1 V& ^' B P# x' f/ ^
Q5:=QuadraticField(-1) ; 1 Q9 D# G d: e, H( i" rQ5; 0 a$ a! I# P5 J. q% S }8 j7 |+ e' G
Q<w> :=PolynomialRing(Q5);Q; 5 U' F/ e6 n# nEquationOrder(Q5);# H y$ g5 s) U2 {4 z: V) M
M:=MaximalOrder(Q5) ; : i+ x& K0 w; Q% zM;9 ]. f6 v! ^& [! a9 y
NumberField(M); 8 R" g5 {! C/ y4 \4 QS1:=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# y/ S* p: N2 j2 TIsQuadratic(Q5);+ v0 ?2 [& {' n/ n+ W* d
IsQuadratic(S1); / J: _7 K; v% UIsQuadratic(S4);# X& p2 _1 Z( g4 E+ W
IsQuadratic(S25);" X4 t* D, g9 S8 a" ]/ c
IsQuadratic(S625888888);9 F: h. x! r& ]9 ]3 }8 y. G( |1 y# `2 Z/ {
Factorization(w^2+1); $ b: M. X/ U( P5 f9 k [8 ?Discriminant(Q5) ; & k* M/ q+ _, {; \; bFundamentalUnit(Q5) ; " h2 i1 j, {' E* sFundamentalUnit(M); / P4 H* l8 w1 [; sConductor(Q5) ;5 `! s$ N( b) ]% @
: c5 w6 o: M- h% pName(M, -1); 9 \) s1 H0 R/ h; Y! j' v8 AConductor(M);; @& R9 b; D0 g
ClassGroup(Q5) ; 0 s, w( k8 x9 B6 D" C
ClassGroup(M); $ o* [) F' l0 r' Y# I, ^ClassNumber(Q5) ;9 M, I1 s% }# t" ^. s5 Y
ClassNumber(M) ;8 P9 G# N, W: V. Y3 M
PicardGroup(M) ; 7 g& D) O2 a! E, WPicardNumber(M) ; # {) Y( _- S/ P4 x7 [ R7 e$ a9 E% g 9 N- r" a: v: B$ m% R$ `" n1 s7 hQuadraticClassGroupTwoPart(Q5);' @5 g7 P1 J4 Z/ S
QuadraticClassGroupTwoPart(M); ) ?' x6 }2 C( \0 {NormEquation(Q5, -1) ;2 o0 Y. P K+ O( G+ A& D$ K/ U% O
NormEquation(M, -1) ;& @! k" C& L. Q' V
% E( z; G) H* d4 e9 j
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field 7 o; e+ O0 j6 n: l) t. j5 }Univariate Polynomial Ring in w over Q56 @, |7 x5 y) R0 g3 C2 F
Equation Order of conductor 1 in Q5 4 W8 Z( O# H: x0 EMaximal Equation Order of Q5 ; L# R) v9 L( l3 Z! UQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field % C+ z: C$ B* |5 y' HOrder of conductor 625888888 in Q5 ) G5 C0 S' O( w9 K0 b6 ] Ltrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field, _! k1 U1 }$ j% v6 ?
true Maximal Equation Order of Q5 ' e& `) @! D# c+ x5 {2 _/ ttrue Order of conductor 1 in Q5 ! e" a! o W8 z7 \! e! Ttrue Order of conductor 1 in Q5% l$ d* [* N! I) J2 S
true Order of conductor 1 in Q5 ( q! g% X/ N1 ?: g[ 1 _- s q5 t0 ~1 l% P' @ <w - Q5.1, 1>,7 @) @- L$ D! s9 D" _
<w + Q5.1, 1> 3 A7 Y/ T0 x/ Y6 O* t]6 o" T# J4 a. L# _; P8 m6 n% U
-4: ]- ~! S& H' P7 m B
, S+ d; E% y8 R( a/ |>> FundamentalUnit(Q5) ; : q/ X! P0 v2 E* ~# W. g8 \ ^, [6 z5 h( [: C/ f
Runtime error in 'FundamentalUnit': Field must have positive discriminant 5 o4 M+ a6 x$ z% X % K) }1 d* e, Y7 O- G4 I3 u+ u+ X! ^; d
>> FundamentalUnit(M); % T o" f+ Y o% v2 B9 m ^ , h }5 i% r( Z) B5 sRuntime error in 'FundamentalUnit': Field must have positive discriminant; Q4 T& R: N/ z( K2 ?7 M" i- h7 `$ ~
% `% B0 Z A: n' W z0 L
4 * s5 R' f. p4 |% I* k$ L* t* b. c0 r , A8 S( v9 C; A4 q8 s/ m>> Name(M, -1); * M5 v# X; F; n% E ^ 8 X, u; T- h) }5 `2 b5 Y) |0 K( bRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1] k, ^& l1 o& G5 x( Z % O- i' i- z! V) }7 k1 q15 y6 A6 s7 Q8 E5 |
Abelian Group of order 1. C; d- {$ A8 n+ G x2 H% \
Mapping from: Abelian Group of order 1 to Set of ideals of M0 r3 t9 H0 u# y$ D8 T" Z/ X
Abelian Group of order 11 N9 W5 a' ^5 x" X! e
Mapping from: Abelian Group of order 1 to Set of ideals of M ! i/ J% p& D* [- v) k. K+ Y v15 X: |* u! l8 z, B- x. z
1 ) J- m8 N) z' \* WAbelian Group of order 19 V- L( \# g9 `# |8 }
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no ; `; K( J" w, c7 v, c0 Sinverse]2 o# M/ t1 y- \' |, Y- |
1 - w! S/ t* x: M# iAbelian Group of order 1) [" c' F0 z i) G) [% W
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant - v7 g: i7 S: t-4 given by a rule [no inverse] % u7 i6 s4 \) U5 E) O. vAbelian Group of order 1 8 C/ s7 _2 w$ ]9 L7 rMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant; j$ ^# F5 t( N' u$ Q B, \8 F
-4 given by a rule [no inverse] 2 m b5 I" X# X) j9 U4 Bfalse : L. _& }, W5 S$ ]- xfalse5 E8 I; A" |% @; T
=============== - u6 }: T1 {2 j. W. A ? ) P0 Q5 ?( k2 IQ5:=QuadraticField(-3) ;% D% k, |( ]( a0 w) ]5 O" h
Q5; " N2 ^9 M6 ^: X ) f+ o& { [7 t* KQ<w> :=PolynomialRing(Q5);Q; - F( s- l- j' y. y. KEquationOrder(Q5);. g- N/ Q$ M) t, d% V
M:=MaximalOrder(Q5) ; ; H' W, d) \. Y6 z8 j" }) s4 y c1 a4 wM;: U5 e" f7 E! q$ N- C& H$ D
NumberField(M);2 e) H: V8 I$ T) S: c$ b
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; " f: S5 V8 I+ x: c2 hIsQuadratic(Q5);" D* O7 s- }0 I1 R6 H( [
IsQuadratic(S1);( Z. f$ I8 P$ ^6 |& V
IsQuadratic(S4); 7 |' C- H* o& _* xIsQuadratic(S25); % j. A1 y* ~3 A# n) CIsQuadratic(S625888888); j5 m! R0 y5 g% E }1 ^4 ~1 Z
Factorization(w^2+3); 3 h/ O3 ^6 v' [& IDiscriminant(Q5) ;, W6 s. F3 M& l( t. J* V
FundamentalUnit(Q5) ;5 ]; \0 _ |- U5 |( Q8 G
FundamentalUnit(M);( v/ c, D9 s( ]9 c' {
Conductor(Q5) ;# s& s' J1 V w& K% a' @
+ P7 N0 ] }7 \
Name(M, -3); + c0 t; J1 T* x* p7 d5 QConductor(M);. T2 O% E( N5 y* D, _* ^7 w
ClassGroup(Q5) ; 0 Z, W) p3 I/ D& B8 \5 bClassGroup(M);7 |/ \/ A& l+ D: L; N
ClassNumber(Q5) ;5 Y) a+ ?' y& U6 y3 N- K8 t, c
ClassNumber(M) ;$ W9 T" I; D- b, s3 b" [
PicardGroup(M) ;5 u+ K9 }8 z8 o/ ?! G4 h
PicardNumber(M) ; 1 k \( C( Y3 a+ ]4 t4 Y/ `: t, Y & Y% u% z2 d0 R: {. zQuadraticClassGroupTwoPart(Q5); 7 a+ B% k# z0 }6 v1 K& \) G. F; V6 zQuadraticClassGroupTwoPart(M); 7 h d. [% c( m3 H9 BNormEquation(Q5, -3) ; # N F, O/ n+ Y# q* q3 F- `NormEquation(M, -3) ; % s. y& W; C2 ^: }- n+ i- C/ J , A1 ~8 C$ |( x! v' gQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field % j! M* b: E0 Z" x$ q6 DUnivariate Polynomial Ring in w over Q5 & @; Q: C+ k, |9 W6 {( oEquation Order of conductor 2 in Q5: I0 W" H$ o8 R# p5 G
Maximal Order of Q56 b) \7 f/ ~7 H
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field 9 \4 t7 I' k1 H* t/ G8 Y& l' `Order of conductor 625888888 in Q53 v% y! R8 n8 Y
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field S1 n- ~9 Y1 C! ]; Y% N
true Maximal Order of Q5 ) j1 `# y+ A8 u- ~8 Wtrue Order of conductor 16 in Q5 & @0 {' r6 H# t- g4 Wtrue Order of conductor 625 in Q5) c9 d8 S( J3 f0 u& o
true Order of conductor 391736900121876544 in Q5 / }6 D' H0 G9 y2 p[" Q1 O. t2 n+ t$ \' \9 C$ c
<w - Q5.1, 1>,( c% F. b3 Z3 `. j- y% ]8 Z% G
<w + Q5.1, 1> 2 {7 I. I$ d6 R# G5 |$ Y]) T" p- G Q. |# n: W+ q1 ]
-3" [& m3 U+ ~* W7 _' Y
6 m: s) o# w+ e0 ?>> FundamentalUnit(Q5) ; 8 e! D) V! W3 ~) l; f0 A q ^ ! _' _: B. F0 n! ?* WRuntime error in 'FundamentalUnit': Field must have positive discriminant ) Q1 D$ S" T' l1 M, b1 F& m' g$ c2 f
" J5 `& a4 p0 I>> FundamentalUnit(M); 6 `4 u; {7 N1 d ^& e- K$ g2 X' i; M; ?2 D
Runtime error in 'FundamentalUnit': Field must have positive discriminant ' S5 T. T( h* \* r$ W- e . ?! j' {, q+ |" x K3 $ W2 L: m( c- P; P: c! {& |# z3 O4 T% ?& q Q
>> Name(M, -3); . J7 G+ L7 h* D2 O3 N% `& k ^ % d" U3 H* L9 U7 i; b8 ~Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1] & z( r! {% d/ c h2 R4 q" |2 }$ y+ `2 K+ A/ W$ e8 D K
1 2 b% ^8 z- R* l; q% e( c6 m# d8 mAbelian Group of order 1 0 P% n4 _" k0 V0 f- AMapping from: Abelian Group of order 1 to Set of ideals of M & [ K/ N8 K' S+ t# @8 qAbelian Group of order 1 ' N7 O# J$ \1 CMapping from: Abelian Group of order 1 to Set of ideals of M 4 ^ d, F' V& \$ x* f3 R1 ! G6 b; a# l2 y% ~% K$ t8 N$ W1) G# I7 h7 |1 {) @3 K7 U6 \, l
Abelian Group of order 1 + g1 R- r O3 z! C9 r) ?; }Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no w' }; D0 F' }9 p6 G7 A
inverse]' g2 B* V0 q8 E6 c( ]
1: {, y! B0 y. W- \# P
Abelian Group of order 1 ! i& w6 n# Q, B! k7 u3 t" E0 NMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant9 M# ^ `) s- f9 p( P
-3 given by a rule [no inverse]" }- }; R3 A/ }' d5 G( D
Abelian Group of order 1 - W8 X' J5 Z9 n3 l) oMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 2 _9 v4 z4 e7 [5 h-3 given by a rule [no inverse]$ O' b/ M5 w6 i3 }; T5 b) h. K
false , b7 _' i V" v, `( Tfalse
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发表于 2012-1-4 17:41
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