. P% w2 w' p, bQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field$ `. P& }0 g0 ` K
Univariate Polynomial Ring in w over Q55 L$ c* h1 A, Y8 H) r/ x
Equation Order of conductor 1 in Q5 , m0 R. I! u0 C* D, DMaximal Equation Order of Q5 8 m" U2 s9 d; y7 QQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field ' R/ l1 U6 G( SOrder of conductor 625888888 in Q5% I* K# S# R K1 M# ]" C
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field: m5 @5 C& ]0 `0 c* P$ w
true Maximal Equation Order of Q5( T. X8 ?. \& z& Q1 m( N1 p
true Order of conductor 1 in Q5 1 w. o- F/ N* q% W* ltrue Order of conductor 1 in Q5 5 ?5 Y4 ^# z- G5 V0 s+ U6 ?8 C7 strue Order of conductor 1 in Q5 2 l; q' b/ J3 f' ^0 Y[+ @4 k- R1 R3 W3 c* c8 b: _
<w - 5*Q5.1, 1>,5 M# M' Q# d+ o% Y& r* a& G& [4 w
<w + 5*Q5.1, 1> . v d! Z! f) d2 ~, M2 A9 n] 3 ?7 S" ]6 h$ E/ P5 T9 J-8 0 R" _& O& e/ `0 {, `& v+ z) u* X5 i: ^% G9 Z4 m4 ]
>> FundamentalUnit(Q5) ;5 ^& v4 f% R4 [' L1 `
^ % a a4 [4 o- v WRuntime error in 'FundamentalUnit': Field must have positive discriminant5 U: Y4 p4 w7 n7 `- U
& r7 t' |* w7 B1 C2 x ( t) ?5 r) j2 |6 \ I# v& k6 F>> FundamentalUnit(M);, N o0 c7 e' O* W: H/ ~5 G
^ 3 t) _% R/ a7 [/ xRuntime error in 'FundamentalUnit': Field must have positive discriminant2 J) z% X. x& h
* p5 L: P9 j' [5 G2 y
8; H0 H3 ]! k" t* X' S
' g6 n- f+ q- x9 k4 h) }2 {>> Name(M, -50);# V2 V) X: Q% W+ j. e
^- W/ j8 W. t2 N, J' x" Z) T# I1 B
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]- i: n9 a6 r+ ]: O
) D1 O4 z7 e; |- T5 I3 q, I
1 ! L5 b& m- E, k1 }2 N: nAbelian Group of order 1 0 ~/ m% B8 D; R* jMapping from: Abelian Group of order 1 to Set of ideals of M; S! h3 J$ t G3 b" d* d- U. W
Abelian Group of order 1, S( p. [. H9 ]
Mapping from: Abelian Group of order 1 to Set of ideals of M' O0 B+ V% C) a0 M0 ]
1! G: N( D) Z' Y: g
19 `, ~# Y% ~# h w& j7 R- f
Abelian Group of order 1. \( _- P! @: x! t4 U: m- ^
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no + k. C1 F+ x5 ?inverse]5 T$ @ J+ r, o1 Y) M: d) ?9 J) h
14 r* l8 q E# ?
Abelian Group of order 1/ h' I3 J% L/ z0 Z
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant/ ~( f l0 A. d; A
-8 given by a rule [no inverse] 5 ^1 m2 i9 ]# l/ bAbelian Group of order 1 ! @, u( j, H4 v4 U7 T2 R0 aMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 2 j4 d% N( `! N. b-8 given by a rule [no inverse] 9 j1 K2 ~% S5 r2 Q4 a: s; i/ `) }false + \" K7 |7 f+ i; z H# F+ Hfalse ( g( B. R$ V1 Q9 ~
看看-1.-3的两种： ( U8 k* R" r5 Y" @# R* T $ F- M! Q0 E9 }8 w) ?Q5:=QuadraticField(-1) ; ! R4 z8 A) G: [1 `: [Q5; 5 v \, N# m/ B; w& r/ v Q7 { J) Y6 `9 K" V
Q<w> :=PolynomialRing(Q5);Q; 7 j* ]! u0 o# T6 AEquationOrder(Q5);5 j9 Q8 |: _( k) T# J
M:=MaximalOrder(Q5) ;. g" z" V+ c' _* Q
M;) @2 f/ N1 ?6 e! g" i8 a6 E
NumberField(M);2 @/ M5 e c) y! N" V) ]
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; P% u: R M) O& jIsQuadratic(Q5); ' I/ S( ]9 y( z/ nIsQuadratic(S1);2 ]- g* b+ S; s! D" L, c5 W4 |( w6 s% |
IsQuadratic(S4);1 f( {) u8 J! o) N
IsQuadratic(S25);0 Q0 b: b) J' R( y
IsQuadratic(S625888888); 2 `# Z, @0 q) m" A n0 d: RFactorization(w^2+1); . E3 s5 v P, g; MDiscriminant(Q5) ;& I2 e2 i3 c6 d1 i/ A( C1 r1 a
FundamentalUnit(Q5) ;: |- ~) @9 d6 ~* E0 s; N" t6 W
FundamentalUnit(M);: G0 i. R( G" q2 o; i, |( Q! k4 B
Conductor(Q5) ; * C! u$ Q8 u4 w9 {1 b) {- K! y S, K1 E- A" F& O5 A3 c. {
Name(M, -1);" M( E6 w9 d2 U+ |, w
Conductor(M);+ c% j i8 m4 q; X! t
ClassGroup(Q5) ; - v! z0 f3 w' R8 V7 B% |# UClassGroup(M); - _) T2 z# M7 AClassNumber(Q5) ; i- I4 k% g/ o1 {" T
ClassNumber(M) ;# @; [& p$ u0 i
PicardGroup(M) ;3 T4 _/ N% a) i h8 E& {% W
PicardNumber(M) ;9 R: Q. x2 v! e+ H
$ _0 |0 @5 i$ P
QuadraticClassGroupTwoPart(Q5); " A& D# ~+ t6 R6 R0 \QuadraticClassGroupTwoPart(M);8 s- O* r; e! m; e; u4 t0 N
NormEquation(Q5, -1) ; " ^& ~" G: s9 E* q. j0 WNormEquation(M, -1) ; 6 S D& e* ^) r* h+ v) P . `1 G& P- ]) t7 [: tQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field . T; e; Q9 f8 Z; @& q) gUnivariate Polynomial Ring in w over Q53 d0 ]3 s/ E( R
Equation Order of conductor 1 in Q51 I9 s. c4 k6 q3 ~& }$ x
Maximal Equation Order of Q5- U z8 Z1 q8 X* i
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field S8 L- V+ S) K5 l$ p# ?- AOrder of conductor 625888888 in Q5 + k V7 _0 P1 Q0 Ftrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field - a% Y& h$ s( z4 a3 Htrue Maximal Equation Order of Q5 - Q% k" R- e: }6 Y' a! ytrue Order of conductor 1 in Q5 1 S1 \% G$ j6 X9 L3 w) Ttrue Order of conductor 1 in Q5 ! q! v& s) J( r6 htrue Order of conductor 1 in Q56 q7 m2 ?( n. ^5 m4 `
[ " A9 A( T! U) u9 |9 P <w - Q5.1, 1>,* T, w0 ~* ^: _: S! h
<w + Q5.1, 1> 0 V" X$ Y3 x# i0 R+ f- u( P] 3 i6 E/ ~& H0 x2 s; L-4 ' p! K( l" U% Q* z2 t1 D5 m4 L) c, V6 H( e w G
>> FundamentalUnit(Q5) ; , a Y$ z: q3 s) C% ~, k o ^ 9 i: p, f; c; [' R' f, m0 p) ORuntime error in 'FundamentalUnit': Field must have positive discriminant" S6 ]( _" P# F+ Q
: x& ]- ]" F8 |& s; b, l+ ` , z& p$ J+ r/ v>> FundamentalUnit(M); l' s) p; t6 ^ ^
^ , U0 S/ N! e9 K# J! d- X1 dRuntime error in 'FundamentalUnit': Field must have positive discriminant' x2 s; p, j+ p) z! A2 _
' F9 E% g/ ?. ?1 u# B+ M4 " a/ `% a* I; x: }6 ]0 `8 j/ V( X: _, L7 C1 F6 C
>> Name(M, -1);8 ?2 j: i) R0 Z2 {! F6 a4 I
^ , z6 t* p% ^8 S! g- L; pRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1] ( |7 [# O: D) |6 Z4 @& a " D; w; S5 [, X m. i% B: `/ M1- ^) T+ T# b, y, f, [2 W1 p. v
Abelian Group of order 12 J* m1 x1 q' ~, \
Mapping from: Abelian Group of order 1 to Set of ideals of M4 {0 [. [& \! G: z
Abelian Group of order 1 k0 V6 C3 _6 L0 c+ H \- IMapping from: Abelian Group of order 1 to Set of ideals of M$ e8 U2 ]& C; Y; a& V! [5 r+ m' {
1 3 Y ]& V1 M9 q8 L3 @1 9 X9 o) z! k$ _2 YAbelian Group of order 1 , J. n& K' x8 J: d- IMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no$ Z- I6 ]9 R4 v8 n( v, c
inverse] + g2 ]! i p9 y& x1 9 D4 V* S# h+ I' HAbelian Group of order 1 4 N! I- F g0 b1 J: gMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant: Y" Q7 _) j) r; Y/ q% D
-4 given by a rule [no inverse]! C6 a9 u; v" u* P7 R/ m* J$ X" \6 J
Abelian Group of order 17 ~( u1 O5 Q1 m6 D! K
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 8 u. ~; [1 _* W+ q" A-4 given by a rule [no inverse] , n! U1 p3 A$ G/ a8 l: L% S4 {6 Cfalse 9 j. y! E' l% G" w8 Y7 L5 Pfalse , \* _# u, y3 @8 A===============: m; o1 L! l1 K* s9 g
4 i& y4 |6 M' ?0 E' `
Q5:=QuadraticField(-3) ; : ^1 r6 T: c% u% g# GQ5;+ c; j5 \7 D3 T- d% U# ^3 ~
0 H' Z8 G* f6 }" h$ C3 [
Q<w> :=PolynomialRing(Q5);Q; , D& b$ D% ^6 l$ B2 ~EquationOrder(Q5);/ C, \, |# ]" V5 |/ b ]
M:=MaximalOrder(Q5) ; # m1 @( `$ i3 h( cM;3 |% E4 z0 J. m$ w2 X
NumberField(M);; |4 ~: U7 g Y
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; ! f0 c4 S, S6 |0 SIsQuadratic(Q5); n5 l- |, ?# t( J. ^/ A, R* @
IsQuadratic(S1); 3 v$ U! |5 M- [. h* M" c+ qIsQuadratic(S4);% p( j: M: }: S% A7 R9 x# V( w+ H
IsQuadratic(S25); " @) D# L6 q! H: eIsQuadratic(S625888888);' Z8 L0 R0 H# l `) y) B1 g
Factorization(w^2+3); ! _& J5 ?6 O2 H6 u: a# ^
Discriminant(Q5) ; ' x6 F c! }, @; g& mFundamentalUnit(Q5) ; , \: h% j" c* K7 o4 c+ M$ S1 JFundamentalUnit(M);/ a) N) }/ o1 t$ a3 v
Conductor(Q5) ;1 s( \0 _7 W$ N& N3 W$ A0 ]6 d7 g+ v; S& @
6 b/ y/ j2 ?4 Y& r; \
Name(M, -3);. G' }) f6 d9 v+ n
Conductor(M); 1 q. h. _8 O1 L. BClassGroup(Q5) ; . k' l. ?) y7 f2 \$ A
ClassGroup(M); ! t! T0 G6 ~6 S5 A1 }. WClassNumber(Q5) ;, }+ n' ]3 k# B& H% [2 {0 O
ClassNumber(M) ; ' p& t2 N6 L$ f) O, H! pPicardGroup(M) ; ' d* v; z- _# ]% c$ D0 mPicardNumber(M) ; 0 H3 K* x c' j! R 2 I. d2 \1 C" DQuadraticClassGroupTwoPart(Q5);$ R8 h+ ?" z2 m G, [
QuadraticClassGroupTwoPart(M); ( }! s. w; i8 R) RNormEquation(Q5, -3) ;) |# f7 M' I, E
NormEquation(M, -3) ;4 f& v2 F0 x) G: t5 Z9 ~
- I" W% M7 b; S
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field/ F1 L2 N1 m& M0 \2 }* V }
Univariate Polynomial Ring in w over Q5. y8 a& S1 s' c7 y
Equation Order of conductor 2 in Q5* c8 n4 e, y" J/ l* m/ u0 ^
Maximal Order of Q5 / J) L: h# V% n$ H, Z5 [, uQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field . l; k; @( u; v, VOrder of conductor 625888888 in Q5. e7 i% ^3 i. }( D! h
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field 4 _1 [4 Q6 N O" [! j5 Itrue Maximal Order of Q5 2 F. P: @7 B1 otrue Order of conductor 16 in Q5& K D9 g$ V1 U1 z
true Order of conductor 625 in Q51 B# ?& k' j! p/ K
true Order of conductor 391736900121876544 in Q5( q, a/ p. W( F3 b/ i
[ - b, h$ h i- [" L5 Z <w - Q5.1, 1>, & W: Y6 X" ?4 J% l @ <w + Q5.1, 1> $ O' T) b9 m5 P& Y/ f- G) `9 g]. c- ~! m1 }4 g2 M! p
-3# x. s* e1 a! a7 P
- z G( U1 G4 M6 o! ?$ e>> FundamentalUnit(Q5) ; 6 [' @# l. a; ]9 h1 {* { ^ ( V9 Y& _8 V' t% f9 G5 C* @Runtime error in 'FundamentalUnit': Field must have positive discriminant! _0 h9 q9 P5 f4 j: r
9 T3 ^7 a: l' `, P( m' g$ b
1 h! }! ^6 ] ]& w
>> FundamentalUnit(M); 7 g3 I: F% k$ ~: ]) s ^ ) i! O! d3 V0 a( kRuntime error in 'FundamentalUnit': Field must have positive discriminant , _- g' S1 j& \ {0 ^" }+ b# J! e6 P4 E) t
3 7 i3 S& V! b0 V2 s5 c; j; r+ v# P; ?5 B/ N6 I5 s+ c' Q, n
>> Name(M, -3); ) E3 ^/ v4 c- P. H; M- D4 r1 y ^; V A3 {5 b$ r; W
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1] ! @3 X( \, g+ ]0 j0 u7 M& Z* q6 w
1 3 I; o* r0 @" NAbelian Group of order 1 1 Z1 V; c6 e$ C- r& j& s$ E/ i# cMapping from: Abelian Group of order 1 to Set of ideals of M - `0 O5 R3 d0 C3 j% ^ q+ zAbelian Group of order 1 4 [2 K1 j; s) C( E1 N; s. ]Mapping from: Abelian Group of order 1 to Set of ideals of M+ s( M2 B* ]7 P4 R5 W6 t" m \* t
1; P9 L" S- L J3 T
1" u8 T5 t: d: e( c
Abelian Group of order 18 i5 j- ? o6 h$ U; Q2 B3 s
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no 2 p5 d8 A# ^& D- @- C& Q" iinverse] 1 H: |; X9 c( `- Y/ D* ?+ \) F1* M% ~& Y9 P: J6 o( p, ?- @, D
Abelian Group of order 1 2 g# X2 R% ]2 r0 [# mMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant2 N+ i$ R. Y. d' Y* ?) v
-3 given by a rule [no inverse]( F, x0 H" P2 {4 i p
Abelian Group of order 1# S- X; d0 D( j2 N+ b' \
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 1 F( [! q7 C2 A7 g- k2 ?-3 given by a rule [no inverse]5 b; p3 K! t9 y. D
false( v( l0 b8 o3 [' n* E% w% C& n3 q D
false
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发表于 2012-1-4 17:41
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