本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 ; k/ q' j9 D/ T( j
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Q5:=QuadraticField(5) ;1 w2 @$ u* P$ R8 K
Q5;: {9 v5 `/ t/ i4 L6 G
Q<w> :=PolynomialRing(Q5);Q;3 r' n2 }3 ^7 J1 b: B+ ^) l
( Y' Q: @" ]7 q! D! Z3 T
EquationOrder(Q5); / B2 Z. \7 T/ ?: L; l @3 c& u: {M:=MaximalOrder(Q5) ;+ m5 D$ k# t' t9 }
M; ) w/ P9 ?8 l2 ~5 C9 g5 QNumberField(M); 9 F2 R c: d& @( H2 IS1:=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 \9 c1 w5 w' k4 i% Q' X. y/ ]! T
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);; g; B# q1 r; Z/ ]/ i
Factorization(w^2-3); 5 e" T' a$ v; J9 p8 [+ f$ o8 M& XDiscriminant(Q5) ;, m. B9 n6 ] q4 [! \ c y, L$ |
FundamentalUnit(Q5) ; 2 N3 o' t' t6 h! w& t4 o' r2 qFundamentalUnit(M); 3 g6 P/ X- J9 L1 k% g: o) M7 lConductor(Q5) ; / o9 b Q$ D$ m, sName(Q5, 1);% g" u; D" L0 u
Name(M, 1);3 ~ ~9 X5 s. Y( k0 J7 `4 t* y+ `
Conductor(M); 7 J' e4 M, Q: Z2 rClassGroup(Q5) ; % |' |4 ?' t1 `) q z) S# j ]ClassGroup(M);& y% A9 a, O% \+ J4 m' ~- C! J
ClassNumber(Q5) ; 7 u0 w5 |- R7 j" u2 ?* R6 y; ?1 _ClassNumber(M) ;* d) g3 S N, ~# R" K
H4 Y( C P+ x
PicardGroup(M) ;4 ]& N- k, P7 B4 X
PicardNumber(M) ; ! J6 M) h5 F, l$ n ]% [) ]+ H. O9 |7 _
( C4 C: ^/ `5 N4 ]& |3 GQuadraticClassGroupTwoPart(Q5); \% K9 x( b# w$ H# qQuadraticClassGroupTwoPart(M); ' l! X9 k( L! }- k 5 I: h9 _# I5 V9 ?: A* ?$ G& o7 Z
NormEquation(Q5, 5) ;7 t2 D6 b1 a2 M Y* t
NormEquation(M, 5) ;6 m% e' _% d: |+ W1 y+ I1 |6 e
1 I" P+ k! S3 h6 c5 R
+ ~3 V1 h: s5 Y5 v' @
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field6 |- z9 o" N: P5 F
Univariate Polynomial Ring in w over Q5 S/ p- m& t6 w, Z3 [* A2 eEquation Order of conductor 2 in Q5- T& d; |& s* D0 z
Maximal Order of Q5 % ^0 m0 r8 u- {9 C8 H3 d7 [Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field+ }/ T2 u8 z2 H2 n( ^) {+ v0 I" H
Order of conductor 625888888 in Q5 9 W8 q7 Q) F, T& g; Ftrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field: m5 H( |6 ]2 s' h( I
true Maximal Order of Q5 & d" J$ Z w9 ~: Q' B/ I. ktrue Order of conductor 16 in Q5 ' o0 Y) h6 s/ btrue Order of conductor 625 in Q51 O6 l3 s9 Y3 E% g
true Order of conductor 391736900121876544 in Q5 : R; z' T* L) W6 ][ % Q% |* \5 q; I) ]. g! C <w^2 - 3, 1>1 G4 ^" z- ]$ d N4 \% W& Q
] 1 I' i8 Y/ u/ C( `- p5 ! E. {: {) l7 l. s1/2*(-Q5.1 + 1)6 C8 v! a; k! ]( _2 _8 f+ }% S
-$.2 + 1 ' l; U* i. V" d( Q. Y5 7 f5 V* j& T. C+ V. yQ5.1 * }" J- r+ x; z5 C- ?1 R$.2' a( \. @4 b8 O
1, `5 x2 q7 f" u0 `( p: g
Abelian Group of order 1 ' z: R8 h) U5 p9 J* uMapping from: Abelian Group of order 1 to Set of ideals of M8 X4 O8 t$ M2 ^: ?7 R0 O: v& f
Abelian Group of order 1 4 K+ u8 }* c8 `( E: P* r% RMapping from: Abelian Group of order 1 to Set of ideals of M 5 @) V; ~' k. F1 0 _: k4 _1 h K) E* c% K, J' D* N1# t6 F4 [! W2 T9 x
Abelian Group of order 1 # t0 Y( c' [( h& b: Z) uMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no ?1 t, G) r- F* m7 W& P) [
inverse] P! j' |& Q& \$ t p, R& k1 ( b F- K; X2 ]1 V% J6 Y0 ]& cAbelian Group of order 1/ P: f* b7 v7 n4 O: R; s7 O
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant # Y0 m1 f5 B" k; B* ?" ?5 given by a rule [no inverse]) U h2 E: x( s
Abelian Group of order 1 3 Z7 o, A" C% V( u" e3 UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 9 T& T/ v: W* `/ j, X- T1 {5 given by a rule [no inverse] : Z+ T- I% \; s$ j+ }true [ 1/2*(Q5.1 + 5) ] 8 H5 L$ F" ~+ Rtrue [ -2*$.2 + 1 ]- [) C: Y9 t1 D% F9 o% m
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Q5:=QuadraticField(50) ;/ G: | b/ b7 Z, Z; A/ ]
Q5;$ |8 ]3 c, r# K! B z2 \; Z
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Q<w> :=PolynomialRing(Q5);Q;" w( N {/ E, _2 M7 w
EquationOrder(Q5);+ F9 V" {3 m; k/ k* R
M:=MaximalOrder(Q5) ;2 S" w" I3 T, d
M;- T4 O% t' w0 R S7 _5 L
NumberField(M);' x- i. g3 a8 Q* r
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;# S: r: b, e3 e6 v# ]$ s. R* S
IsQuadratic(Q5); : }* z( k1 M( _+ ~' ?' y1 V1 uIsQuadratic(S1); 4 U9 g5 f5 K5 L H. g8 R! c+ f$ FIsQuadratic(S4); 8 C/ m9 [3 g( {$ b' \( `3 L7 TIsQuadratic(S25); 2 B: H- C* j; [- l6 `$ r' ]* L6 F2 DIsQuadratic(S625888888); ; w- x. a. J8 M" ^1 [ nFactorization(w^2-50); ( [% s D9 z+ Q) t. FDiscriminant(Q5) ;3 ~1 z$ @- x7 P# d! a
FundamentalUnit(Q5) ; ) L. S( V9 b- P5 y, |FundamentalUnit(M);6 } Z' m3 m" c9 h% x" V! a
Conductor(Q5) ; b @5 A- `# M1 l$ f x$ Z; W$ \( h- r" W3 y
Name(M, 50);: U, m' m! L$ @" h9 `0 X. Z. T
Conductor(M);$ i! R/ H* a. c' g0 Y
ClassGroup(Q5) ; $ _/ M+ I! `4 HClassGroup(M); ( Y) ?# ?! h% NClassNumber(Q5) ; ' @4 S8 r4 O: b- _+ ^1 i5 j% h. RClassNumber(M) ;8 x5 u8 O+ ^1 L$ {1 w/ a* x- }
PicardGroup(M) ;& q8 g* Z+ c+ g
PicardNumber(M) ;3 ]; u& `3 y% O6 L
j& S" q- V7 I8 i
QuadraticClassGroupTwoPart(Q5); - D. W+ t' v2 K- k, z* C3 kQuadraticClassGroupTwoPart(M); 2 {# M8 d4 E- [4 dNormEquation(Q5, 50) ;/ K, h6 A5 p9 r- L. U
NormEquation(M, 50) ; ) V% v* Y! N3 q' a; G2 B3 [: }& C6 G3 E2 @* f5 L
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field0 G! M1 F% X/ G; l" m4 Z. Y6 _( ?* I
Univariate Polynomial Ring in w over Q5 : f3 \5 \4 g' q/ DEquation Order of conductor 1 in Q5 : Y4 l/ t; y6 o/ X* U6 c; G r7 P, NMaximal Equation Order of Q56 a; o L) {% r8 ^) M3 y
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field7 A" }' W2 d- A/ ]& N5 ^9 t! p
Order of conductor 625888888 in Q5 3 g0 I. x. n. k6 S) i) htrue Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 5 o. j( e: A7 B* Z& qtrue Maximal Equation Order of Q5 ' y( Q! h d; X etrue Order of conductor 1 in Q50 P( E0 S7 N" [
true Order of conductor 1 in Q5 2 F h8 a9 M' N" |) S9 o1 g Btrue Order of conductor 1 in Q5 # G& B8 U6 Z: [/ y8 H: q( o$ h[5 C5 Z+ L: O9 c" I7 p
<w - 5*Q5.1, 1>, # t. q1 C+ Q: A5 \ <w + 5*Q5.1, 1> 6 Z( l; |! M) l8 B] & O2 S+ n: T" R+ j& Q9 n8 : \( k0 `# ^9 z/ a. p3 M$ kQ5.1 + 1 . X9 Q+ R. A3 ^; J7 u1 a, v( Z$.2 + 1, [5 H2 D/ t2 A+ X
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>> Name(M, 50);( G/ d: {% X$ `6 m& x9 W0 g
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Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1] ; s& }+ a( K( T! V7 z4 R9 d % x/ D; r4 h$ r# y: b1 _/ W2 y0 e4 T6 Z H, D' d* O& CAbelian Group of order 1" h# B; x) Q8 h: F5 B" b8 N
Mapping from: Abelian Group of order 1 to Set of ideals of M ' J+ ^* X' o8 J9 R$ P& ]6 KAbelian Group of order 15 h9 P2 K/ g0 j& K' Y! x* u
Mapping from: Abelian Group of order 1 to Set of ideals of M; V* Q" w4 a0 `& n* t8 b$ H
1 ) K+ F7 v8 j+ u- Q6 R2 ]1% a. Y V6 ^% X$ V+ s; D Z
Abelian Group of order 1 5 F6 w* s& W3 {2 E. Z+ h7 YMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no% K& c/ D' w2 x! t
inverse]/ J n0 \7 g3 Z- T! K
1" A5 G- ^- m2 b" Z- O
Abelian Group of order 1 o5 K" T1 a9 d4 u _' a \
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 6 L7 A( x& L* q+ t& Q! t8 given by a rule [no inverse] 1 }8 ]; g% W# [Abelian Group of order 13 e: m4 l$ c) N( Y
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant6 H2 q% V; L, h$ Z
8 given by a rule [no inverse], z5 |* I9 |( c. G
true [ 5*Q5.1 + 10 ]" e; a5 J" J! [2 q* K
true [ -5*$.2 ]