本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 & h& b+ r. n) w' {: b# q, H. f+ O) i
Q5:=QuadraticField(5) ; 5 R0 @- d+ b% [Q5; 2 Y4 {2 t! `! r/ {& Z! OQ<w> :=PolynomialRing(Q5);Q; 6 f+ p$ C0 {1 o# `/ n4 H( O 4 E! g% D8 V' \0 V5 jEquationOrder(Q5); * M3 J; s& _# z sM:=MaximalOrder(Q5) ;& r q v6 s" E$ c5 p8 l' x
M;3 j; D1 Y$ }% S
NumberField(M); % F# P$ i Y" U- z) e ^9 @9 YS1:=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: i/ e9 ]% T; V5 hIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888); # {+ Z. _1 ? W7 k, J$ l lFactorization(w^2-3);) B: R. g! \, m( @ a i
Discriminant(Q5) ; 3 t# n* u V1 ~FundamentalUnit(Q5) ; - o8 j" e! k+ d. l# I* `FundamentalUnit(M);0 R9 Q' i4 T# N- e! M; E
Conductor(Q5) ; 0 }6 A, j- r8 D3 m" g+ r8 HName(Q5, 1); : o$ w! b7 n8 H% A% J* R$ D- cName(M, 1); 4 ?6 y7 `$ w+ v0 {' i2 t2 aConductor(M); ' d% D3 R$ c3 p2 a: c% C% H, A) F; sClassGroup(Q5) ; # n Y0 s9 ^! M" f8 _0 i! Q. c# w+ hClassGroup(M); 3 M7 C V7 y$ ]5 p8 U- Q- FClassNumber(Q5) ;" n" a0 G, G! B5 ~( X
ClassNumber(M) ; 1 b% W1 X8 P+ \ ' u# k* z8 x1 C5 A" wPicardGroup(M) ; / ^* C4 [1 B6 S2 g( r6 E7 ~PicardNumber(M) ; ' b) D2 B; Q, R 4 ]" ^ u. {+ _. U' g% E/ i; e( o! J o( x" i/ E' q
QuadraticClassGroupTwoPart(Q5); ; @9 a; Q: U- t- p) ]4 p2 |QuadraticClassGroupTwoPart(M);* Y: E4 E: K, U( L
3 Z+ `& L' W$ }& m$ h5 Y; ?/ V/ z / j+ s& f: g( D3 V6 { v$ ?NormEquation(Q5, 5) ; " ?# l* d4 c' r1 x- z: @ \NormEquation(M, 5) ; ) r6 G3 ~/ M+ d1 o5 h" a U d* G0 U7 w E; t( s3 x3 p% Q; R& l7 P0 F; q) q+ H
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field: G" g3 X" \( ~# b
Univariate Polynomial Ring in w over Q5 5 n8 @" z( [. o" m* d; g" VEquation Order of conductor 2 in Q5- J- b0 `1 c$ `4 b# Q
Maximal Order of Q5 ( M8 ^9 K7 j' r* M8 g$ X5 n' Y2 PQuadratic Field with defining polynomial $.1^2 - 5 over the Rational Field8 t7 y6 [3 Z* M; y
Order of conductor 625888888 in Q5 $ o2 o% \& W' ~) X' @* r* ?6 W1 xtrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field ; @% q& V9 m* z* j, @, ^8 z8 p$ wtrue Maximal Order of Q5 , q( ]6 I# T/ |4 J1 ]$ qtrue Order of conductor 16 in Q5 5 H8 L! c1 a2 S% ~0 r& p- Qtrue Order of conductor 625 in Q5+ Y. a$ m+ {1 u! x+ O: V
true Order of conductor 391736900121876544 in Q54 m; j9 K6 T' N; j1 J5 g1 ^8 {3 @
[ - y% ?: l ]* i5 F- A( i) _1 w <w^2 - 3, 1> - I1 o; k1 G8 _ G+ }" |! u8 S] # R J5 |" P8 y1 w4 _6 X L I5 ; u) o3 G' z h! Q6 S0 }9 g. E1 z1/2*(-Q5.1 + 1) 0 S V$ d- y0 A& f, ~4 V-$.2 + 1& [- l2 p% q" z
5; @9 F3 g" u: h, G8 B) O, a
Q5.1 : H! a/ V! b! ^$.2: {: W$ L. O; m' E, Q
1+ W: v5 Z. t( w0 j% J, o! b
Abelian Group of order 17 U6 p) H: c0 p9 F1 \* O9 L
Mapping from: Abelian Group of order 1 to Set of ideals of M/ r& Q8 F1 ~7 X
Abelian Group of order 17 h4 j( p' B" G( {0 t
Mapping from: Abelian Group of order 1 to Set of ideals of M/ C0 P: H) Z1 u' ^
1* ^# S# t: y; o8 H' _$ v1 D
1 2 O5 M, P! d* Y$ {+ Q1 `6 QAbelian Group of order 1 ) p6 F: f0 }0 _; @3 s/ ?1 l: }Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no8 L2 B( v$ [) I) _) Z" O
inverse]3 f& J. b# ]. h/ R4 V: z( h& F
1( |, j/ Y$ ~/ B4 p4 }2 p
Abelian Group of order 1# o9 t& a5 ?, C9 l" ]
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 8 I% l* A( u: T @3 k% o5 given by a rule [no inverse]* A; @. a% N1 Q& i( Y0 h
Abelian Group of order 1 ( O: `! c8 W2 G) kMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant8 W0 H9 c [+ c2 ~# ^3 H6 c
5 given by a rule [no inverse]# U l3 j, f) T9 l) C
true [ 1/2*(Q5.1 + 5) ]. v. K# N5 I! s* z) S0 a
true [ -2*$.2 + 1 ] D8 @ p1 e4 }2 g( R' \5 i; Y
# z' x* O2 ~3 y# m$ E( D4 l 1 L7 E& L4 c+ w) t! i0 P4 B# j* E: {7 m, \0 _
& u# j' i$ U+ {4 T: P' M
% x$ x$ f J( F( o {7 H/ v2 S9 L* J' N
; I! @/ }; e2 d
' i7 g+ E# g1 Q& ~4 V# N 6 w/ v. K* Z, W' z 9 w* s/ D+ Q* ]) l2 n - }% _) n* F/ h' b( | f; T7 k. K- a============== . w8 A3 e: x$ x% F& P5 i8 B; |( Z' F$ F
Q5:=QuadraticField(50) ; ) g4 K- d+ O4 {" xQ5;3 N2 S& `3 t/ _7 z
0 S* b: g8 }* f
Q<w> :=PolynomialRing(Q5);Q;3 Y, I1 ?5 O; o: p" z0 Q7 w
EquationOrder(Q5);1 ?- J! v# e1 x! V! G# p
M:=MaximalOrder(Q5) ;2 J; R" o) S4 U; |, u* _" [
M; / ]7 B7 j9 O' G8 gNumberField(M); ! z- m% x R B0 J2 {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; / B7 q4 l% r2 E( CIsQuadratic(Q5);* g) B6 {' Z: v9 _ @
IsQuadratic(S1);7 N$ Y. y: \! J6 Q9 o5 N
IsQuadratic(S4); " G# b2 j# W0 v1 P' b* V* g; kIsQuadratic(S25);2 J3 s, A+ i5 H x0 t
IsQuadratic(S625888888);; M; y" a& e8 x2 D
Factorization(w^2-50); 0 P+ u) W0 J- T$ g4 YDiscriminant(Q5) ;) S7 J! s3 [; p! N5 x
FundamentalUnit(Q5) ; ! k& V8 m2 `8 N. FFundamentalUnit(M);5 N P! r" O: `5 e
Conductor(Q5) ;- p9 h' d2 \; N5 y* c2 B( K
% X' s/ ?4 M0 \% FName(M, 50); 0 E0 ~& I O% gConductor(M); " X( z1 P `* O: eClassGroup(Q5) ; 0 s* P Y. |' J& G2 `7 S9 m9 @ClassGroup(M); ; y' _- g/ ~: T" uClassNumber(Q5) ;. ^. }4 O! O# q. e
ClassNumber(M) ;# d% P" U+ M, }& S l3 ]+ {
PicardGroup(M) ;. o/ a+ Y) ]5 k K
PicardNumber(M) ;1 H( k3 n: h+ E, @
) k$ R" r3 U5 |& C" YQuadraticClassGroupTwoPart(Q5); + J- K8 }' \- L, lQuadraticClassGroupTwoPart(M); \. y& ]' ]2 d a6 NNormEquation(Q5, 50) ;) L8 U5 S% i" A5 W7 S
NormEquation(M, 50) ;& r9 g' o0 N1 b: b+ W8 w3 `+ N
. g! J2 O1 h. \: a
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field ! w& Y/ K. U* ~Univariate Polynomial Ring in w over Q5 6 J3 G3 N" A2 V `Equation Order of conductor 1 in Q5 $ I6 B' U; w" B/ I0 U) QMaximal Equation Order of Q5* Y7 e, d) M2 N& [( H n9 m& y3 \
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field1 y: k5 h, I+ y# U/ B; O" B, X* e0 B
Order of conductor 625888888 in Q5 5 ?1 X, v+ v8 A% h; _true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field" {. Q4 j/ B0 n: Q- q
true Maximal Equation Order of Q5 2 C! N/ y {" E/ |2 t: I5 atrue Order of conductor 1 in Q55 ~8 p0 A- W/ \5 u# \( s# Q
true Order of conductor 1 in Q50 P+ b( D2 v$ \% G+ h2 j
true Order of conductor 1 in Q5 ; f& @0 v+ m; j: N$ @% B[/ B* W( g3 V3 y
<w - 5*Q5.1, 1>,5 j5 R0 |9 ^" ^" X/ `% J
<w + 5*Q5.1, 1>( G7 F/ Z$ b) X' u, k0 R
] 0 [2 S- H. t; L% e8 h! u# \; l8 g1 r7 B: F% a8 @Q5.1 + 1# {. O$ Z8 Y% y8 R0 u& j
$.2 + 1# C: t& Q7 C8 P9 O
8/ L( E$ O1 D4 T7 p3 k6 l
& e3 t0 A7 a7 y5 a. c" n
>> Name(M, 50); 7 F$ o! Q8 [: M j ^ E) [" K) `4 ]( R! LRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]3 K( j& x, B0 v b: N
4 w, ?5 X- v# y! n8 e
1 0 p* Z, l6 c- cAbelian Group of order 1 , D. N R/ L5 _! p) K! ~' Q% {Mapping from: Abelian Group of order 1 to Set of ideals of M - `. |* U9 y$ T; e0 s/ _1 x- Z) pAbelian Group of order 1 # G6 Q- D" K! ^$ r* V! u: X: \Mapping from: Abelian Group of order 1 to Set of ideals of M5 n1 [8 h9 O3 T
1 + v1 @: i' \, f0 m. `16 u m3 _% M# B3 h0 P
Abelian Group of order 1% `; @7 H8 V2 b
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no) l$ W6 T2 ^' g$ w
inverse]3 G; l' w+ A' D, ~
1 , T; @$ e4 S# t' cAbelian Group of order 1( L. O6 G% r( ~3 J, X8 T
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant : E! |' K: U, P+ L8 given by a rule [no inverse]+ F. a Z8 ]* m% \; L4 b
Abelian Group of order 1$ r" C2 G& o* C( e3 `9 K* B' c
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant4 R" p5 t& u5 J8 n. M# U9 J
8 given by a rule [no inverse], w9 [4 H0 o) x( |2 |5 u
true [ 5*Q5.1 + 10 ] 8 }" M, Q! ]! ?& U& x' r2 ]true [ -5*$.2 ]