本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 + S% m5 |; b9 W, T) a" v t $ ?+ n$ v0 W# X a7 [Q5:=QuadraticField(-5) ;: ]; r3 i0 i$ ~# s/ z- i+ @% [
Q5;; j X5 C+ s" a& B j
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Q<w> :=PolynomialRing(Q5);Q;1 d( O& g, s% c$ Y
EquationOrder(Q5); & t+ }% j5 j, E$ r% ]M:=MaximalOrder(Q5) ; ) a, g3 V5 \' ^5 t8 xM; & D9 Y. k: e- R: v8 f) zNumberField(M); 5 m3 |# J* Q( h5 w1 o- v1 JS1:=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; 0 o) N6 q" H; i7 s5 ?2 o2 r, MIsQuadratic(Q5);# P& p+ l, x7 ~) k
IsQuadratic(S1);. Z6 f( x# i2 J+ A0 f% U2 f# ^; I
IsQuadratic(S4); ; P, I3 y! N2 _$ z6 q, `3 MIsQuadratic(S25);) O9 B' O+ U' D8 J8 P2 Y& C7 u
IsQuadratic(S625888888); , U1 X7 L ~9 \; D& s$ yFactorization(w^2+5); - L5 S* p( |5 D% }' D1 Z" ^Discriminant(Q5) ;- K! R/ u8 K. p; P, @1 v: q
FundamentalUnit(Q5) ;9 a6 N$ f. s! C2 M1 P2 L
FundamentalUnit(M);! q X- p7 p5 f8 k7 D# M
Conductor(Q5) ; # P0 V. B+ c$ S( h2 f8 f) m. Z3 d) T/ Q0 s9 F
Name(M, -5);# l9 h. v* j0 u ^
Conductor(M); % h2 Z6 N( I: V8 t; aClassGroup(Q5) ; g: R/ N3 n- N9 m9 F
ClassGroup(M); ! R3 A) H# r; \. G6 z$ {% K ]ClassNumber(Q5) ;5 Y) k+ w& N; J4 U# M8 F
ClassNumber(M) ;' L) u) k, Z, Y5 a0 I9 J
PicardGroup(M) ;9 x& ~7 `0 q- N* @5 c
PicardNumber(M) ; , o$ C- z' W) D4 h7 _$ O ) M. A) {: p& D& _QuadraticClassGroupTwoPart(Q5);( D! y- x- u! G* K( v
QuadraticClassGroupTwoPart(M);- }8 O( u0 Y4 r: l
NormEquation(Q5, -5) ;0 N) ?2 }2 j! I& j
NormEquation(M, -5) ; ! r/ i, j6 n' i" zQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field6 T8 S* j" @3 m w8 [' X
Univariate Polynomial Ring in w over Q5 8 h8 e5 a# c% z) D6 [Equation Order of conductor 1 in Q5 * U4 q3 }$ C9 Y n3 @$ m1 t. ]Maximal Equation Order of Q5 ' n; r* O' {2 b+ `* X. a( v' }Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field ( ?" O3 P3 \" W% EOrder of conductor 625888888 in Q5 ' } |* s% P7 L) {( _true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field ( \* T' S0 ]; r, |3 Atrue Maximal Equation Order of Q5/ o' F5 m( H% q+ E+ o% j- r) K
true Order of conductor 1 in Q5' |; M% Y2 R. U
true Order of conductor 1 in Q5 3 T% x: E0 K1 E q! otrue Order of conductor 1 in Q52 h8 q3 c) P. f5 J# \$ M
[ 9 s8 N J+ s1 t6 g# G/ J- q <w - Q5.1, 1>, $ D6 K$ B' Y9 {% [! R! w, s0 { <w + Q5.1, 1>' A( {4 Z9 a, r
] 0 d' V+ g% S! [" R3 I6 P5 m-20 ) [% c) y T; \" \; d + ]+ v: [4 F$ _* \ _7 e>> FundamentalUnit(Q5) ;/ M* @ b/ J5 I! W7 J. _
^ 4 f0 l" p2 e0 B- }1 T, x2 X" ^5 bRuntime error in 'FundamentalUnit': Field must have positive discriminant 0 _4 O3 c% Q. n& {' Z' l# Q) T. U& B3 B# `) w& E% M- i0 p+ W9 ]
+ y% }8 g: k& j2 D>> FundamentalUnit(M);: r6 P8 g+ { ?1 ~/ p j% Z0 h$ z9 I
^8 a, h. N4 E2 O' b8 q
Runtime error in 'FundamentalUnit': Field must have positive discriminant 6 z6 H! j' z/ T7 T% Z3 K & j& G9 I% f. u20 . p1 s; L/ m" A7 o + `+ k0 c. `5 i" o) a# Z>> Name(M, -5);! t. F z, T' n
^ - S* q: T/ f S. p; ?, L. ~- pRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1] + q% K" K+ C7 h6 U' g# x- | 7 ]" Y8 ?) a0 u; ~2 [! N D1 , s! J9 L5 m3 W, \& l# dAbelian Group isomorphic to Z/20 K6 y! ` H/ g+ V" G/ b
Defined on 1 generator 2 N; T, R/ Z: ] R$ dRelations:4 Y% ?" n1 U% ^2 u- e& Q
2*$.1 = 0 6 ?8 D. ~, e0 Z* H; M! MMapping from: Abelian Group isomorphic to Z/2* c+ v! X; {' B4 L. ? h
Defined on 1 generator. {) P1 s* j/ B3 L
Relations:/ L( L' Y: }) t1 q3 X* `5 O. L6 R
2*$.1 = 0 to Set of ideals of M 2 V& y6 f) e4 \. d- E, t# o: Q& PAbelian Group isomorphic to Z/2 ) t# Z0 Z3 j& Y; e- c. D, dDefined on 1 generator' X( |) Q: D k& D8 Q0 k$ m
Relations: & m+ Y2 }2 Y# L1 k* x1 F. K 2*$.1 = 0; }" }, t5 {7 V+ Z1 I5 K F
Mapping from: Abelian Group isomorphic to Z/2! `$ F. t, a& z$ J' M' o
Defined on 1 generator " d' M. V9 n5 xRelations: 6 k2 w4 E4 g* J7 Q7 j 2*$.1 = 0 to Set of ideals of M 4 s. r. o; S/ ~) C' X, |9 y" Z20 {" ^- \, `$ x) E5 u9 E" s
2; s6 z9 o7 @" p! E3 L, q
Abelian Group isomorphic to Z/2- V, e3 F1 h5 P1 B+ |
Defined on 1 generator; l2 I8 B& R+ j% G& a5 {. `
Relations: 6 h/ {3 T! v4 @6 Y* E 2*$.1 = 0 # a% s8 U* B; ]& K% e2 r. uMapping from: Abelian Group isomorphic to Z/2 . D1 u& B# s$ k1 m4 jDefined on 1 generator # T9 G9 V: A; dRelations:* Y3 ^: U* ~" l9 q* C$ v" h
2*$.1 = 0 to Set of ideals of M given by a rule [no inverse] 1 f2 x2 u* ^" d: P: n2 I. x2 . E9 t. G3 c6 I7 X- i' OAbelian Group isomorphic to Z/2 5 R5 A& K; I$ q7 pDefined on 1 generator0 t" e+ q- _6 W4 ~% U4 C
Relations: ; d K2 |8 F' Z5 {/ ^/ _. [) K* v 2*$.1 = 0 . v( T- v" e/ [9 O5 p7 jMapping from: Abelian Group isomorphic to Z/27 A& @. w' N4 D+ T! {
Defined on 1 generator ( Z2 E L ^; \5 j: h" GRelations: , Z* v; ~; ?- Q4 c2 h 2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no 1 V! e& f' ?; |9 A& u; finverse] 5 f9 c: Y) i6 J. S( b3 U+ _5 b( EAbelian Group isomorphic to Z/2) R+ D% h" N. ^3 A6 }2 L( \! @
Defined on 1 generator- Z: o; J; l* A7 H6 I" ~1 i
Relations:# V1 L/ Z* V0 P, t2 V
2*$.1 = 0 ) r2 k# Z8 ]- r' }8 O& V1 P; tMapping from: Abelian Group isomorphic to Z/2 0 g- D" g: D7 m% q: oDefined on 1 generator " \8 x% C+ C. T# i hRelations: ; c8 z% ?3 e6 r# O: z 2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no 7 r) Z. f6 r" v7 Hinverse] , Q4 u' ?2 D8 ^4 t( ^$ \2 efalse ; f2 l2 R5 d3 v+ O& mfalse% L, ^2 q/ Q4 J: P- D; m
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5 o1 F; R7 U# B' o- g0 R2 G) EQ5:=QuadraticField(-50) ; 5 y5 b& ~# g5 BQ5;/ x8 S* i- |" Y& I
2 V: _1 X; V2 T# N$ w
Q<w> :=PolynomialRing(Q5);Q;2 j/ B% S. N% y+ B
EquationOrder(Q5); |& b: C8 }; z) |$ A/ h
M:=MaximalOrder(Q5) ; ' l# ~: ^9 Y# v( A& w. _5 Z8 u: VM; ' K4 y2 ?' q7 \) w, x2 B1 ONumberField(M); & P; I+ u9 j5 U: B3 [2 g1 }4 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; o, s3 k0 F0 w1 |$ N* \IsQuadratic(Q5); 3 O" @) x& d/ P" XIsQuadratic(S1); ) c( ~( _2 W9 Z/ x4 r5 sIsQuadratic(S4);3 \2 m" r x6 U/ d- ]3 ]
IsQuadratic(S25);7 i: ?# u6 B& y9 y* u
IsQuadratic(S625888888); 6 P7 d5 f2 H" U( d7 ]2 C6 ^) W; jFactorization(w^2+50); $ V% a1 v5 C) [: p9 W
Discriminant(Q5) ;4 d( q8 u) Z& H; p7 Z; i2 _
FundamentalUnit(Q5) ;3 h0 E- G f' P1 Q# y
FundamentalUnit(M);; p7 G: T1 |+ q+ r7 _
Conductor(Q5) ; , j6 \. D1 I. Y) T0 a & W( j, f; Z4 T+ |( Q3 k. r, S: n/ ]* ~Name(M, -50); : s: }, S7 B1 K/ l" UConductor(M);2 B- G- |/ a- S' i
ClassGroup(Q5) ; / V' z& i2 _. E+ _. ]ClassGroup(M);( R6 b3 F, {; B ^5 I; H; G9 g# `
ClassNumber(Q5) ;" v1 J4 Y( v, [- |
ClassNumber(M) ;+ R& S5 w3 Z2 F4 l) [5 M
PicardGroup(M) ;7 B1 @1 c; e2 R! b$ z. V
PicardNumber(M) ; 8 T7 C t. u, p$ ]4 l3 U4 o6 k) S+ t: e# w* ^1 ]4 [/ J1 W
QuadraticClassGroupTwoPart(Q5); % n4 h4 U3 i" sQuadraticClassGroupTwoPart(M);; n; i: [ M% x& G4 E) |% ]
NormEquation(Q5, -50) ; 3 N- N. w0 @3 s) X& ~NormEquation(M, -50) ;* Q" D' X t3 G- |4 J7 i
, N. ?% m+ }$ ]; n( }- ~8 V8 I0 QQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field/ F9 E" G$ x8 | d3 w9 k2 L% v3 ~
Univariate Polynomial Ring in w over Q5; b& r% |9 C8 [
Equation Order of conductor 1 in Q5) L D% C" W8 Z* h1 F
Maximal Equation Order of Q5& }8 g' X- c0 F A: c
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field0 J: W3 m' }6 T) t
Order of conductor 625888888 in Q5 . `( r8 T0 A: O3 c+ @8 _$ _true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field- K/ ]* a7 V- _ E% v) q) d) u" _
true Maximal Equation Order of Q5 5 f( r* H- A- f0 E" Utrue Order of conductor 1 in Q54 E# a# Z3 S2 X$ U
true Order of conductor 1 in Q5. ~0 B# ]: r0 V- f5 ?
true Order of conductor 1 in Q51 [+ E! F5 \* G. V# {
[! y9 L. v) h O- w* h8 ]6 Q
<w - 5*Q5.1, 1>,! m9 D! }9 ~& F5 w' M
<w + 5*Q5.1, 1> ; @5 l, ?, w, e1 ^, x$ e7 u]+ L7 S* T, n }# a$ u( R2 j
-8 : I n9 c% m9 w# D6 y , C6 x) e4 W/ t- t& g/ m$ k: T0 R>> FundamentalUnit(Q5) ; 9 T$ `. v! y: G4 x( v& u8 N ^ 3 ]# {* V o; e8 C4 Y4 J9 h# lRuntime error in 'FundamentalUnit': Field must have positive discriminant/ V# o+ G* ^# Y: j
$ T+ S1 Z4 J0 k3 o6 Y/ Q! x . P! v, s; t: n3 S>> FundamentalUnit(M); 9 k0 a7 {9 D8 n0 `% U ^* ~) N G; ^+ J F/ t6 d* j0 ]
Runtime error in 'FundamentalUnit': Field must have positive discriminant; i5 d% V0 h( Y, I7 E# C* z
* f" i( u- d' V6 b+ j# X4 P9 p$ L8; ]3 D" p. n( R8 \# u0 t
7 Z3 h' X, B4 f6 q! \9 G7 T( [, y
>> Name(M, -50); @% s3 p$ h! Y4 T K
^ 1 Z! F4 n# A: @$ G' rRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1] ( B/ O5 o# P2 A0 F/ v' I! x1 c. k' U' o1 }( k+ U8 I8 y
1 5 t' E9 @/ e* Q tAbelian Group of order 18 ]' P& O# S% ?% g
Mapping from: Abelian Group of order 1 to Set of ideals of M1 w) t* O6 v5 n+ ^) ^) }
Abelian Group of order 14 l/ o, V" \+ c5 H6 Q
Mapping from: Abelian Group of order 1 to Set of ideals of M' @& w4 _' {4 ]4 _4 e# R
1 $ y& j- m( J" f( u9 R3 c1 1 t2 F1 b9 q! w8 B A- _& {Abelian Group of order 1 9 Z. e& M5 ~* }/ R0 }. fMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no 6 ]& R/ K# c* ninverse]! ^5 d1 L* V F6 P9 O3 G
1 , T8 ^1 S) `0 m/ S( f' M$ dAbelian Group of order 1" k( c! ]9 h! G3 X+ W8 f+ a
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant. t! m. F5 N/ a v
-8 given by a rule [no inverse]7 X. E3 D' Z; g0 s% i
Abelian Group of order 1; `9 ?: `0 ^& r) `, ?9 i( F
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ) j. l. `) S7 v2 t) R5 O9 {-8 given by a rule [no inverse]6 f0 x" X, a# k$ ?6 g
false / ^) q% S5 b; P. Ufalse; q& T6 \! v5 l) w' ^8 p
看看-1.-3的两种： % H0 f% `- o# r. r4 U$ M. h . h3 v5 ?/ ^7 L, QQ5:=QuadraticField(-1) ;% l; s3 | B, ?9 F
Q5; $ N# K$ \: ~; h4 ^' ]+ Y$ c) S0 ?, ]7 C8 ?
Q<w> :=PolynomialRing(Q5);Q; : E4 O9 s" q# c6 u9 {& jEquationOrder(Q5);; N% T) [: {4 \
M:=MaximalOrder(Q5) ; \/ Z- R, U. Z: M) l- }6 b: |M;/ l$ |; x$ _# n1 X4 X
NumberField(M); " N K m& n8 w3 o$ `. ^- _. u* T" US1:=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; ) @. y: G3 f( C/ u6 w' j2 uIsQuadratic(Q5); ! j' n$ ]6 s3 }0 \+ ?3 h9 l( YIsQuadratic(S1); 5 N- v- @7 a( O# U9 LIsQuadratic(S4); . U; v6 D' L1 o) fIsQuadratic(S25); }8 n6 Q0 b: x% D( L" E
IsQuadratic(S625888888); \( }. a' s- n" T. m7 [% ?/ @Factorization(w^2+1); 6 L1 a5 k# s" A* a r W
Discriminant(Q5) ; ; ~1 G$ t, L; ~4 q( WFundamentalUnit(Q5) ;; R/ V# w, v/ I, `) O
FundamentalUnit(M);) a' u$ {3 P" _) M
Conductor(Q5) ;) l: B6 j2 q4 f- }7 d! J2 u
" S3 ]4 d- b, kName(M, -1); & _" I' i6 c; R: P5 [, BConductor(M);5 Q& T( h+ V1 T. R( x8 b( ]
ClassGroup(Q5) ; + o. S1 Q6 j1 L; MClassGroup(M);5 _1 Z3 F# Q- d2 `
ClassNumber(Q5) ; ! J3 G% {7 ]- ]8 |) o9 x* R) QClassNumber(M) ;5 A" k+ l' Z& \
PicardGroup(M) ; 0 A; Y# l) T0 _7 ^/ E6 Q( A6 @PicardNumber(M) ;) {9 I, v$ p( N \/ O! z
. h# C( U3 q2 w
QuadraticClassGroupTwoPart(Q5);6 [- Z. o: _0 V' J5 Y
QuadraticClassGroupTwoPart(M);9 Y! q7 E. \( m2 N1 j* o* q1 D
NormEquation(Q5, -1) ;: G9 `- M0 }+ i/ u) I' E
NormEquation(M, -1) ; 8 `9 ^ ^4 A$ ~: K, T 0 L- o$ g. {: t& \; g) b! aQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field4 }# ^- M: z" S- c' K
Univariate Polynomial Ring in w over Q5 . j% c. D U; _& FEquation Order of conductor 1 in Q5 u2 x$ L* r2 _, } F
Maximal Equation Order of Q5 $ y% ^1 E1 X( ~9 M; s, D% yQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field/ f, r: Q% K+ M9 t/ @
Order of conductor 625888888 in Q5 2 b. ?! Z9 z( [/ ]! }" Ztrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field' Q, ^. ?; |, }1 C/ {, H
true Maximal Equation Order of Q5% v M) u X' O/ Z0 k
true Order of conductor 1 in Q5 - W5 b0 d$ N( c. b' y( _ y6 itrue Order of conductor 1 in Q5 5 u1 F& _2 q' D6 `1 ttrue Order of conductor 1 in Q5 6 F. g! Z2 ^0 G4 M[, A3 j& C% u- @: f& X% C- j
<w - Q5.1, 1>,, B D0 E* a1 e' s
<w + Q5.1, 1> 8 i8 H- d/ U) c# N] 8 }# [7 U& ]- G1 t( k-40 d6 t4 `9 s1 g
+ [3 |4 Z+ |2 `2 o5 W
>> FundamentalUnit(Q5) ;- C4 K1 w0 @9 d
^ - l8 X1 D# T! e5 q4 SRuntime error in 'FundamentalUnit': Field must have positive discriminant 1 S: c/ d7 W3 P0 Z. P2 T 6 `( c" a0 }4 \" k. m: W4 o: j9 @/ J) y
>> FundamentalUnit(M);" m5 v5 S* B/ j+ w& J
^7 B7 y V/ R8 l2 L8 G! J5 M$ R
Runtime error in 'FundamentalUnit': Field must have positive discriminant8 l& ~0 d; m2 C' z( {5 `
/ R$ |6 k# G! a- ?) ~5 H4 @5 O* C42 \% j$ q3 D6 Y5 o2 b: Y% W/ b
( W- |: z9 l0 J. [% L
>> Name(M, -1); : x7 d7 M6 i& t5 f- y; W N ^7 L' m( N9 g" _# e3 M
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]4 x' e" T1 Y; `9 W3 T& V
+ N( h; L2 a' J! t/ A1" R% o* R2 S9 N* \; m: E4 Q7 r+ o
Abelian Group of order 1- n) G- A8 X# l9 Y: s& _% X
Mapping from: Abelian Group of order 1 to Set of ideals of M " f' z! W% ~' k8 vAbelian Group of order 11 C8 S+ j% q9 T8 W5 y+ o; {
Mapping from: Abelian Group of order 1 to Set of ideals of M8 v, Q, {4 w' X$ u7 ]) r) V' V
1 {- i& }% G5 Y
1 : W; a& w6 G0 c w# XAbelian Group of order 1! t9 n/ `3 R4 d |& |* l0 }* F7 L
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no1 d0 E8 P ~) Z7 W
inverse]* J( N* |3 i# U4 y8 D" c
1 * `7 Y( Y! Z: Q- y# v) M- f' q/ {Abelian Group of order 1" `: ~3 F3 @8 z X1 ]2 p F
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 1 V" R- Z' y& s' K5 \5 e) Z' c, v! M-4 given by a rule [no inverse] ; d- h" V/ X! r |Abelian Group of order 1 + J, y, l1 n0 }: Z8 qMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant # i. d, T& V6 n8 s6 U-4 given by a rule [no inverse]" \& `* `6 D4 c3 [& t
false" x) B) A$ H# D% Y) I$ E0 D) s
false7 q1 f: t: G2 {4 D9 ^8 F0 @* l
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0 ?( }3 ^8 S8 e; k/ yQ5:=QuadraticField(-3) ; ) p8 _* V* h# r7 z/ HQ5;% u0 v) U9 @+ b% |+ e$ u) Y
: M2 Z$ }/ Q6 [* W1 v
Q<w> :=PolynomialRing(Q5);Q; ) N" E8 m4 J2 k7 b7 D: qEquationOrder(Q5); : Z8 L% s# P6 v# }( OM:=MaximalOrder(Q5) ; 4 c9 r+ b8 K2 I+ Q) K; q, \M;! H' N4 L, {: z
NumberField(M); 8 g3 s/ Z. D8 L, Z- NS1:=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;1 j- F2 y7 f+ ?6 G1 \( Q' h
IsQuadratic(Q5); 3 X; o9 l' I$ I2 I- v cIsQuadratic(S1); ( z' [0 L9 x7 `! NIsQuadratic(S4); 0 t6 x$ O0 I; C4 W# J( y) DIsQuadratic(S25);" k, k" k6 j1 t: G7 Q$ ~- ~
IsQuadratic(S625888888);$ |! ^7 {2 c: U
Factorization(w^2+3); & B6 F' }, u( w5 `. A2 M5 t. DDiscriminant(Q5) ;8 P' Y: }- q" M1 m, b2 Q/ p# v
FundamentalUnit(Q5) ; $ h. y7 u8 e: UFundamentalUnit(M);! R1 g* o7 W, f# _
Conductor(Q5) ;( A: @, G6 J7 N/ P( h* @2 ~. C
4 E* R4 l5 ]7 L: ~. |Name(M, -3); / c/ E( B4 Y* o* M1 ZConductor(M); ; f( m# @5 z2 J5 mClassGroup(Q5) ; ) d' U1 D/ H) E. CClassGroup(M); " S7 X- r. s, a+ LClassNumber(Q5) ;0 S4 f- n2 \9 |) b
ClassNumber(M) ; " }; F; t5 K" MPicardGroup(M) ;7 O; a" H1 z" x* `
PicardNumber(M) ; P: ^4 H, N0 z$ i: R, z $ V2 ^4 j2 j( Z' p8 D. @" G6 z' YQuadraticClassGroupTwoPart(Q5);9 L1 k# L3 ?6 c
QuadraticClassGroupTwoPart(M);! E# u4 Z* P; z6 s6 L8 a
NormEquation(Q5, -3) ; # @1 M8 b9 Q: ?6 H+ @NormEquation(M, -3) ; 0 e0 u; v8 W# v, q 8 ^4 d J. y2 [- CQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field0 ^+ n' R5 \7 T* U* m W
Univariate Polynomial Ring in w over Q59 q$ D, f, k8 t/ |0 z
Equation Order of conductor 2 in Q5 + S7 `0 a8 q2 ^5 jMaximal Order of Q56 Q9 F, G' f+ ^% T
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field 8 G) X. p6 h$ Z/ \5 uOrder of conductor 625888888 in Q5 / B0 K2 X. }: G, Z7 Ntrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field 9 P0 b: w' t3 U) f& H0 Mtrue Maximal Order of Q5- \, Q$ q4 R* [5 Z, w1 c* t, |2 |
true Order of conductor 16 in Q58 I% A5 r& u4 F8 s$ O
true Order of conductor 625 in Q59 [( D0 p& j5 `' c
true Order of conductor 391736900121876544 in Q56 Y% S4 K" \3 v- @* \+ m
[# p& \9 {8 L; `1 h9 l4 I
<w - Q5.1, 1>, 1 v2 ~3 r1 _1 e! t% a <w + Q5.1, 1>; r7 a m, p0 O. ^) \
] ; `( N9 w3 j6 j2 b4 X4 \-3 $ \' G" |2 }6 S0 ^2 `0 l, Q- @6 m/ l' W7 `1 u+ n6 L$ I
>> FundamentalUnit(Q5) ; " a1 b7 c3 l9 M$ _; ?* I- v* G% C ^ 9 z. M5 l& o$ F0 ^4 pRuntime error in 'FundamentalUnit': Field must have positive discriminant ; d+ B }/ b6 k3 t7 ^% T* J Y' v( H: V
1 d# |. a5 |1 J. e7 C; k1 W$ g3 Z _
>> FundamentalUnit(M); / [8 q& g6 U4 d$ l ^( ^* T- g& r& F* k f
Runtime error in 'FundamentalUnit': Field must have positive discriminant ' u; ^+ ]# O I. C# y2 D 3 H4 k% z7 Z; v3 ]3* l- t$ u9 ^" ?
0 v4 p4 L4 G; @) k) u9 @>> Name(M, -3); 8 ]% n: v# m: Z$ I6 s' ^ ^1 @: ^, `$ {0 i2 X5 s+ Y" g
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1] , m& T; g4 P6 y; v# V8 a8 H ) S9 o. p' Q6 s# t% n8 B1. `) H J( R: m6 E' q
Abelian Group of order 1. d& L) t+ ` F8 n+ ^" v' F. {
Mapping from: Abelian Group of order 1 to Set of ideals of M- ?- k7 J, b, y4 Y7 D
Abelian Group of order 1 % [0 A4 p6 C7 Q* j) |0 z. ]+ U/ vMapping from: Abelian Group of order 1 to Set of ideals of M0 \$ l% H8 w$ w* |4 N
1 6 @2 W/ i, W' P X! J15 i: a( l2 Z8 g; E/ c, c
Abelian Group of order 12 R# K" {5 j& B9 A2 D; u
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no) z: O: s/ K4 U/ h/ Z
inverse] . V# @3 e- L, q6 _7 `' D, h3 M1; b' g3 d/ f I* z( J# [) y
Abelian Group of order 1 ( Q1 x) L) O7 aMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant : W5 `$ y8 L/ ?; V# D) P-3 given by a rule [no inverse]+ e1 k, B+ P: G* T7 n; T) g2 O$ y
Abelian Group of order 1 " ] I9 Y0 @; P+ a: I' {Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 4 T( k3 ^) ]2 w: M/ a# V& u1 }-3 given by a rule [no inverse]! `; [, c# m7 [7 M( ~' p4 u4 U
false ! s; f* {: I- ~+ q& gfalse
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发表于 2012-1-4 17:41
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