本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 * V* ^: b, y: P' H3 ? * V4 u6 w3 g9 aQ5:=QuadraticField(-5) ; . j9 S( F( y7 w' |, `8 H) `; ^# VQ5;: E: s* @* I9 v4 p2 u
7 K) h( u+ m/ D0 ~9 w# n
Q<w> :=PolynomialRing(Q5);Q;7 z/ @; g4 F4 s& u8 S
EquationOrder(Q5);6 D2 Z- @$ q0 z4 P4 h; a, I
M:=MaximalOrder(Q5) ; ( h6 a* u* {6 v3 c) [M; ' W: t" g) H& h1 z3 s( M# ZNumberField(M); 2 ]6 u9 y3 R( c8 y$ M! j8 D( 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; 9 n- s6 `" N2 ]; o8 tIsQuadratic(Q5); 1 y" P5 |9 W! I! S5 f! t% F EIsQuadratic(S1); 1 C# W: l( T9 I; z6 U/ A1 ]IsQuadratic(S4); ! \4 m3 v5 }$ y0 J6 E# hIsQuadratic(S25);* o$ f) v6 |& n' g( U
IsQuadratic(S625888888);$ a1 o8 M5 a6 |
Factorization(w^2+5); y! E/ B( l' B( g6 K7 B- w, N1 f
Discriminant(Q5) ;, B# @% d, Z2 N. v4 A
FundamentalUnit(Q5) ; , S" R" B( X. A3 h0 DFundamentalUnit(M);5 V1 T' G1 f A$ m& [) K9 V
Conductor(Q5) ; # Z& J) I- s1 M$ O* t( B2 A ; E. v9 Y, i! C' z$ E6 g5 cName(M, -5);( `( A) m6 a, a
Conductor(M);+ t# ~7 c, L: B: r1 ^7 q
ClassGroup(Q5) ; ( _) e) c8 y( O1 o# a
ClassGroup(M);* o# r) z3 Q2 A0 h+ o* g$ F- J6 u5 J
ClassNumber(Q5) ; , k6 Y5 @! b1 n9 i- |+ y) |) [ClassNumber(M) ;5 E# N7 f4 T5 T% H J
PicardGroup(M) ;9 c. l" s2 h5 w
PicardNumber(M) ; ; L1 ~+ E R, J9 u6 M! ^: K% t- E3 {$ B* Z/ a W5 p& i
QuadraticClassGroupTwoPart(Q5); ( y, k0 b# Z% @0 mQuadraticClassGroupTwoPart(M);' z; W% M& m6 D O* Z! H
NormEquation(Q5, -5) ;! V5 z4 |2 i3 j( h- w5 T
NormEquation(M, -5) ;/ W; D3 u+ v, k8 G
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field 1 v* x1 i" P7 e8 y8 T% [) R1 xUnivariate Polynomial Ring in w over Q5 5 K4 |8 U" p# n! F2 G* }- Q; LEquation Order of conductor 1 in Q5 / t6 |7 e$ z7 N' i9 AMaximal Equation Order of Q5 , i& I; f+ }6 Z# YQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field 9 ]* t, E/ F: ?% e1 X* D5 }Order of conductor 625888888 in Q5 * J }( z' b) ^true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field . `5 |: l& X. c( @/ _true Maximal Equation Order of Q5 # w1 d# Y% G# q2 c G: ptrue Order of conductor 1 in Q54 m9 T# f& x3 {% N
true Order of conductor 1 in Q5 & E* p/ r4 L8 k+ f8 Vtrue Order of conductor 1 in Q5 e @1 T2 B ][2 N- T0 R! \6 ?& c8 I) O6 n9 l
<w - Q5.1, 1>, $ ]' b5 ]# A+ S* V) }2 q" b1 m <w + Q5.1, 1> ) M# J6 d" Y* E9 K6 g1 [. p: I]) e2 U( L+ a b% i
-20 " F2 W# [: H7 l" _" _. r3 W/ z3 j 1 x2 Q5 _; J* L/ y" ]2 J/ E>> FundamentalUnit(Q5) ;/ `3 q9 ?. n. X3 t. D. ]# n ]
^! k+ j% J* m+ q$ r/ ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant * ]. }' ]0 W+ z, A 7 }* Z. E2 B8 F/ b& p& k! r3 p/ l& D( z5 R2 e. b# c2 a0 e. q
>> FundamentalUnit(M); + j0 W0 L0 C( g& d5 q ^9 H j. X8 s& t7 @3 M) T ?
Runtime error in 'FundamentalUnit': Field must have positive discriminant " Z4 c. ^* I% n, I( y' @+ a, j( M" d4 ^
20 4 Q/ W, v5 }* [0 G& q- }2 W* ?+ C3 X1 N2 C2 m
>> Name(M, -5); . V9 w2 J4 g8 R' }+ P" C* w& y ^) T S7 @; e& A
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1] : g) T. l$ }( ]6 g; d, f1 a6 R) Y! U* I" R; v
1) \$ y) ~, a, t+ w' ^
Abelian Group isomorphic to Z/2 ' C& ~0 ]" V0 ~% p% z) ]Defined on 1 generator! v* G2 S& }- w P ^$ \- h
Relations:% n, J. k% M- k- U* X
2*$.1 = 0 * M; z# R8 Y; NMapping from: Abelian Group isomorphic to Z/2 ( M7 W: u8 X- S2 CDefined on 1 generator: S1 J% ^- ?; A+ e/ d8 T. G
Relations:0 e4 N7 y7 d' h& F6 o
2*$.1 = 0 to Set of ideals of M0 I& Y+ R+ `5 S" k/ ]: Q1 c2 v
Abelian Group isomorphic to Z/2% \/ D! h! m. a
Defined on 1 generator / F& }) K" W' j/ m3 p7 a5 {0 mRelations:( L( n p5 p# i7 ~% D
2*$.1 = 0 # Q. a5 M$ B1 XMapping from: Abelian Group isomorphic to Z/2" ~; V- D5 Z( o# q9 a: {
Defined on 1 generator" ^% C( [/ E! m3 u( a/ v
Relations:! S% N3 a, a; X% _" M" ?
2*$.1 = 0 to Set of ideals of M C* C$ J1 U6 i7 ]
2; h3 ^, h9 I/ i8 i% P' A
2 . R# H E4 k5 a4 h1 B4 tAbelian Group isomorphic to Z/2 6 X1 h- \; T) q2 A/ Z0 [Defined on 1 generator' j2 I+ u' s# _' d) M$ [
Relations: 1 f8 B9 \$ _4 g. i 2*$.1 = 0 3 z6 F* h$ F3 t# {9 e0 N% J# \Mapping from: Abelian Group isomorphic to Z/2 2 x6 `# C, g( wDefined on 1 generator 0 g( d; Y7 T7 b: a. n' H6 z1 ~Relations:" f% y2 |0 U/ q- p4 ~. {" @
2*$.1 = 0 to Set of ideals of M given by a rule [no inverse] 2 X# d8 u! `0 {2: S1 j) J. v5 O" L3 B
Abelian Group isomorphic to Z/2 4 [# _9 R* S! ~" v! K& u3 FDefined on 1 generator( } T2 |9 Q4 a5 C
Relations: 7 N) t8 ^- |1 p' C/ F x 2*$.1 = 0, X+ Q8 E; { Y3 U9 [9 \
Mapping from: Abelian Group isomorphic to Z/2 " J: _# B/ Y/ B' s4 p0 xDefined on 1 generator" l- f& Q& b1 C8 d ~
Relations: 5 V% k8 e' D: v+ A, \- g! ` 2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no ) R" d5 p. M- u1 G% U
inverse] 8 v% D0 z B5 E y# C2 OAbelian Group isomorphic to Z/29 J( r, b$ H5 Z+ K" K* w, [
Defined on 1 generator0 ~4 z, w: m% @1 F( N3 j8 b# d
Relations: ; U% C4 Y' i" h- t9 P 2*$.1 = 0 6 o. Q+ s8 n* Y' c! J* eMapping from: Abelian Group isomorphic to Z/22 u9 Z$ Z; I3 M$ i8 b4 n( L
Defined on 1 generator# f3 w% |3 U# l3 X' k. s2 e7 f$ M
Relations: - a8 M) Y S7 `: u 2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no ) u' f/ u u/ l. finverse], z. `( }; U( z2 Z% e4 Z
false $ @, w1 _! \5 V! yfalse 5 l/ M/ t; |4 ~# J============== |* K' M9 f8 O
+ `- W- _" } D1 p2 E6 W/ O% a4 k) g" e
8 p' M1 \% k4 p% N8 ]: D1 S" c) K, eQ5:=QuadraticField(-50) ;* q6 W" D9 y/ V; U) V C4 ?
Q5;5 Z! P; K4 E( y! A' R) V7 W
$ a7 |- c' d! b) D+ p
Q<w> :=PolynomialRing(Q5);Q;% g9 ^; e6 Z# n1 o9 B" x
EquationOrder(Q5); 0 C9 R. L" ]" j' {* ?M:=MaximalOrder(Q5) ;) h: \$ O2 J+ z: ?1 N
M; ' B4 v2 O0 T- z' C" @NumberField(M);2 g& ?1 ^- _0 c6 Q
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& S' U4 c& c# }( D6 L' xIsQuadratic(Q5);6 [2 s5 m) \5 I+ \4 i6 V
IsQuadratic(S1); ) F) a% D; a. `IsQuadratic(S4); 0 c& O' i0 F/ U2 X; uIsQuadratic(S25); / P/ Q- E9 e9 EIsQuadratic(S625888888); 4 b, q: ]! o+ N0 T) r( {5 L3 RFactorization(w^2+50); 9 Y$ b" v( |: P7 W, l7 v8 TDiscriminant(Q5) ;* P4 O! x P/ M: p( V! }. S
FundamentalUnit(Q5) ; + z" S, o3 X6 u/ r' t( }FundamentalUnit(M); # Y) ?4 Y6 F4 ]2 S8 B! ?Conductor(Q5) ; , q! v/ T0 X6 E7 u* L6 z5 `7 A6 i3 k5 T$ H9 Y" p% L& d b
Name(M, -50);& x. W. w- Q% C- u
Conductor(M); : P! o9 M2 O v5 N1 a- j% Z- e2 MClassGroup(Q5) ; 4 e1 g9 b! P! WClassGroup(M); - L/ R5 j* B% H" s( qClassNumber(Q5) ;; s( l# G. j4 v; m8 w
ClassNumber(M) ;, \' h, h; v2 ?( |6 Q/ W y( @
PicardGroup(M) ;- r1 w+ a5 |9 r5 }' M
PicardNumber(M) ; - B# k2 Y/ @ h, @+ j& x8 N ' ^. U$ }2 J! r$ d. XQuadraticClassGroupTwoPart(Q5);3 M# T: k% G/ ]7 z0 d6 K. e; W3 N
QuadraticClassGroupTwoPart(M); 8 u( A# h' V+ c9 T* ]! TNormEquation(Q5, -50) ;4 t0 q5 ^0 S% H6 @+ w
NormEquation(M, -50) ; : ?" ]: d: G" ?7 ?7 c0 T }5 M$ Q, x% k) IQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field - g6 U2 r w% V: I1 Z2 f; y, xUnivariate Polynomial Ring in w over Q5* [' z5 E: ]7 z; h, C l4 i
Equation Order of conductor 1 in Q5. e, R/ s. I' K: g
Maximal Equation Order of Q5 6 N. t6 @2 S8 D/ vQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field / ?/ _. q, h3 h2 l, ?Order of conductor 625888888 in Q5 F, X# W0 A8 B! U1 x
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field. i. x- V: ^+ p1 v% C" g8 y, d1 g. R
true Maximal Equation Order of Q5' }/ Y; f0 C" u
true Order of conductor 1 in Q5# K9 A1 M* o) L6 ^1 f
true Order of conductor 1 in Q5 - Y6 C2 c4 L% B) H4 s' {true Order of conductor 1 in Q52 D: f( I' E: h! a m6 f& L. [
[! ~% @# G. U. n* T, ?3 E+ ^/ A9 O
<w - 5*Q5.1, 1>, 7 J7 z1 v- j! [% B7 ~; ~' { <w + 5*Q5.1, 1> 7 j. c* W- r2 V]+ P) v) {: K/ P6 l( P4 [' g3 o# ^3 c
-8( }' i- W& |6 D5 U3 t1 m% Z1 V
- v! Y) R" u# ^6 `! F# K>> FundamentalUnit(Q5) ;* V/ _- V* `. B2 U% e! E
^: |$ \! C( t3 D3 s' q) ]. N! O
Runtime error in 'FundamentalUnit': Field must have positive discriminant9 J9 o9 P! D7 D( m9 E4 f
) ^8 H3 x$ I& i/ K* C- D0 d$ y! S5 {5 b; z: R* [% y2 ~
>> FundamentalUnit(M); ( r! o. H/ B# x8 A ^ 9 I1 e+ H1 N0 t( L6 ]Runtime error in 'FundamentalUnit': Field must have positive discriminant0 W. X& D5 h4 s* v# e
: n0 |: O. [, g) b
8 5 ?+ f W s! W/ z+ p& ?0 K: P) D: U+ s. c: Z& r
>> Name(M, -50); U, k& m; L6 A: c( g1 x
^ , I" U( Y$ p+ ~; k: nRuntime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1] 4 c+ C( M4 w T; ?: O3 \0 J* s ! a9 N2 Z3 J5 v4 F3 B: d3 P% r1 . ^- f3 {! L% V0 S+ L6 c$ m- JAbelian Group of order 1 / U9 I% B8 z) s K3 _Mapping from: Abelian Group of order 1 to Set of ideals of M ! f5 \2 j& X# M# e$ LAbelian Group of order 1 * j2 b7 U8 V: s" [Mapping from: Abelian Group of order 1 to Set of ideals of M: r) l7 X, X2 `+ l. ?6 m0 x% v* z
1$ ?/ N% F- S3 O
14 R/ h1 p, i2 w7 x
Abelian Group of order 1 % |. }# p+ j+ ^6 N7 p) PMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no 2 M; E/ q0 ]: N e9 z# W+ Zinverse] $ x/ P2 m4 r3 g# x& |2 y6 w3 W1' Z3 @% Q, R7 a" j3 \
Abelian Group of order 1 $ V9 ~6 h9 W3 c/ `# D0 zMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant" f; y/ {' K! k1 N1 y+ _& Z3 E
-8 given by a rule [no inverse] , j$ n( C; a; KAbelian Group of order 1 & Y& G7 `8 F4 D3 y8 v5 I6 q" zMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant " R3 g3 {( {0 _8 @" G- ?7 S& ]0 g-8 given by a rule [no inverse]. y! O. Q- e3 _+ C
false' x- l* i8 j5 P3 D3 c' ?" A" v4 e
false 7 f' \5 ~1 F8 j& ~3 D z
! u8 o- E' W" D. Y: EQ5:=QuadraticField(-1) ; 6 J0 d4 R; G1 l3 `( o4 Q/ j3 E) NQ5;' e. x, {1 H9 ?! H3 M
T& t) \) I/ e6 E- O6 ]$ d
Q<w> :=PolynomialRing(Q5);Q; ! T/ W$ `6 s; A) B2 O/ D3 yEquationOrder(Q5);/ }0 O0 X) j; _- ]% ?
M:=MaximalOrder(Q5) ; # I( V8 y/ }- V; R! B5 n3 U/ @# YM;& m; Z2 u" a3 R" C
NumberField(M); . u$ r9 _1 H9 y2 w* {' p5 GS1:=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 d% d3 \, F# X
IsQuadratic(Q5);! i6 T) m3 T, x' Y9 l5 q4 v
IsQuadratic(S1);# g+ w" t3 E- V: _2 v
IsQuadratic(S4); " u9 T& V, i* }! N& Z) eIsQuadratic(S25); 5 c' g+ I3 L1 C7 [9 s6 ?" p% R$ G) fIsQuadratic(S625888888);. ?4 S! p( j, z4 [ A0 z
Factorization(w^2+1); ; K4 [ ]' r5 d9 D% M, g$ h9 l& qDiscriminant(Q5) ; ! V8 ?: c9 H3 u$ q% w, lFundamentalUnit(Q5) ;8 A+ H/ A) |5 d$ i0 {4 {
FundamentalUnit(M);3 o" D' a6 }; A7 X7 P4 D# |
Conductor(Q5) ; # [' o0 |5 ~' `5 S5 g8 x 8 j1 z" Q) c8 Q9 ^5 F1 x! S7 DName(M, -1); Z5 s1 F4 g2 I. n+ EConductor(M); / M8 `. r" C% e( T) d6 e ~' C G. Z9 CClassGroup(Q5) ; 7 E# Y5 h3 u) a5 s9 B
ClassGroup(M); . y$ E8 X2 I* ^, yClassNumber(Q5) ; 8 H9 ]: j i8 ~, \% Q2 _ClassNumber(M) ; _- c3 S. O9 A1 xPicardGroup(M) ;5 s8 p& `$ W( T* U- C" I6 M0 F& s
PicardNumber(M) ;# @% j% Z8 E/ O- v! W: L3 k
2 K* {8 }( A# p- w! DQuadraticClassGroupTwoPart(Q5); 0 [" @8 W* C# n% O0 lQuadraticClassGroupTwoPart(M);0 ]% D" e2 i: l1 r
NormEquation(Q5, -1) ; 8 G& a7 Z1 @7 i; z5 k' z: h) yNormEquation(M, -1) ; ( f7 \2 ^' s# ?8 S( e ^& c# z$ F) C$ I: hQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field f1 \, V3 } i, rUnivariate Polynomial Ring in w over Q5) x# k3 J) o7 _4 M9 h
Equation Order of conductor 1 in Q5$ U0 e, J; I& L M m
Maximal Equation Order of Q5 - h& y2 [9 T6 Q& O w ?Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field( C4 ]2 y& u0 L
Order of conductor 625888888 in Q5 + M0 @: @% ]3 F k, o- B! ^true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field& N; |" `4 ]. X {+ G6 L4 O. B6 x5 G
true Maximal Equation Order of Q5 8 o k1 i$ j. x5 Strue Order of conductor 1 in Q5 : ?# _! w4 c, [# a Ntrue Order of conductor 1 in Q5 q; y9 c6 i* htrue Order of conductor 1 in Q5 j$ a" H5 R6 x' O, S4 ]% r, M[' G& _& R1 [4 [
<w - Q5.1, 1>,7 \9 {' H- e* g2 G' k9 B
<w + Q5.1, 1> $ |1 ^) z1 Y; u) m0 V( J8 b]8 O J+ ?- d/ R; u! ]' g2 ^
-4 & ? B# ?. P7 q5 r$ ~/ Q |: a7 X6 d, F
>> FundamentalUnit(Q5) ; , Z4 s: T' ]5 v: y* j* I ^2 A( A8 s' i, e$ _) S6 t0 H( a
Runtime error in 'FundamentalUnit': Field must have positive discriminant , Z5 I- {+ m( C- f. T8 U; w: G' i( V) w
- K" N" k ]! A) n ^1 B>> FundamentalUnit(M); 3 R: m2 L: p$ w/ ^! D4 t' e3 U7 B ^: X/ G, w6 r6 x8 Y( T/ q
Runtime error in 'FundamentalUnit': Field must have positive discriminant# ~4 l4 N8 Z# [
1 I$ n" J( R1 p; f g% B4 ( H* y7 J0 b1 p9 D' E8 a / [( |) I2 q- O9 h; S>> Name(M, -1);4 T: a/ u# B7 |
^% ]* e/ C& e% e
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]& Y& A* w) z, J! u6 E% @
' k; f9 Z. O: [' X+ Y5 |* n1 , B+ b* ]2 b D6 KAbelian Group of order 1 8 T Q- |# ]# r/ N6 d& d1 X+ R {2 ]Mapping from: Abelian Group of order 1 to Set of ideals of M, U* a2 A, x4 B/ ^$ K) ^( h# B
Abelian Group of order 1 9 T, h/ Z8 |& J b2 I& aMapping from: Abelian Group of order 1 to Set of ideals of M. e |0 _! z. Z* Q q
1 u/ B0 u& z' D9 P; d1 k* u1 / T6 w" @' R* B' F7 Z4 NAbelian Group of order 18 l1 y- f. y( ?
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no T' h% b: f7 |" `9 |0 Uinverse] - J9 M" ^5 u5 _7 p, {. a1' b$ V- ?; M9 X. C: }! {/ J% A- N
Abelian Group of order 1 # J" k2 T @! }5 R7 O, RMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant* \* C' `- z7 O' H
-4 given by a rule [no inverse] ! a9 [, Q" o9 I7 z) ~4 T. fAbelian Group of order 1) S: w7 ~% |8 i- o1 i+ r) o
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 7 Z% R3 P* |+ b3 \. q3 |( y/ o-4 given by a rule [no inverse]7 l! S% w3 _3 g
false e& L9 X4 ]! K" m' _& O* c6 i7 hfalse2 g( I8 b5 _& A- w5 ~
===============/ i' s3 k+ J7 Z+ o
5 N6 d- Z, X( F3 ~: Q
Q5:=QuadraticField(-3) ; * v) I+ {: a8 B1 M0 H m" aQ5;7 C! ~6 c- q, F
/ R2 M+ r* f* S8 v9 g
Q<w> :=PolynomialRing(Q5);Q;$ G( J O# K& ]5 M& I ~$ L+ f) A' t: k
EquationOrder(Q5);& o f; J& D' b' o! E" J' J
M:=MaximalOrder(Q5) ; 7 n& x& z9 c7 o M! u! S$ X- `M;5 S/ G! X. u r8 o, u2 `
NumberField(M); 4 n4 K5 B; _5 xS1:=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; * j% M' }6 V s4 K- F1 ?IsQuadratic(Q5); I$ @& a* i% yIsQuadratic(S1); 8 [3 G. E1 b' lIsQuadratic(S4); 5 ]& f3 ^$ Z$ | v P' wIsQuadratic(S25);1 t8 s4 X* w% p- s* V
IsQuadratic(S625888888);) D% {( L# g& a6 O( @
Factorization(w^2+3); & h( N% H) j0 X! B0 N
Discriminant(Q5) ; * Z0 m" @. P; N0 R7 `+ GFundamentalUnit(Q5) ;4 Z" O5 R3 e9 x) @* P
FundamentalUnit(M);2 B7 e, e+ o: F
Conductor(Q5) ;. \0 ]7 X8 b% e/ ]. X! J+ x2 M q
' N3 W+ k, `' n, ~9 ?4 F# ?Name(M, -3);, o% Q9 y, N: R* `
Conductor(M); 7 }5 v) v8 F5 k1 v- jClassGroup(Q5) ; 9 q3 {4 W) J6 h4 t' d& Y6 xClassGroup(M);) G, g7 _4 ~% c
ClassNumber(Q5) ; 3 q, A2 a& i- c; v6 }ClassNumber(M) ; 0 F4 w- v) K2 ZPicardGroup(M) ; 1 k) f; c4 h5 _% o0 MPicardNumber(M) ;4 R X" h Q- ~* c- @( V e3 H5 _* R
. _. h# s; t3 S8 U- @) J% x
QuadraticClassGroupTwoPart(Q5);/ ?: c1 p( Y% g7 b2 ]# v: `9 `4 L
QuadraticClassGroupTwoPart(M);( k4 R1 A/ `9 q# S
NormEquation(Q5, -3) ; # b; d1 S( }9 Z: f8 |: @6 q" HNormEquation(M, -3) ;5 H) I9 B P7 s$ h! H2 {! D
5 ]8 l& Q4 ]- B9 ]) WQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field ) j8 Y5 Q# T9 xUnivariate Polynomial Ring in w over Q5 ' N5 d% ~+ T, Y4 B V `Equation Order of conductor 2 in Q54 c" H1 F: {2 y/ @
Maximal Order of Q5 6 V. K$ g. r7 VQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field' Y7 ]- ]' ?; v. B$ j
Order of conductor 625888888 in Q5 " C( U F$ J& o5 Q+ dtrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field 7 y7 K- P0 W1 z# K* ^true Maximal Order of Q58 K5 Z F+ C( f! s2 o! n
true Order of conductor 16 in Q5 ; O. `9 V, q% t( M) e% p* [& ztrue Order of conductor 625 in Q5) w- a7 o$ e; J* D
true Order of conductor 391736900121876544 in Q5/ x- L) U; i" d% J
[ * b# P6 O0 t' \* y7 T' m <w - Q5.1, 1>, : V7 h ?; q( d* i- o <w + Q5.1, 1> & V9 w* z: w3 U% r( G' I5 u1 `: m. V]' Z" B X( T- k4 Q# b
-3/ ?5 f! g9 b/ w$ t: w
9 t5 }" z) n. }/ @7 n
>> FundamentalUnit(Q5) ;" S* S4 b6 |# i
^ # D7 ? Y) m( r/ [$ ^Runtime error in 'FundamentalUnit': Field must have positive discriminant " ?7 i+ h% h0 d9 Y , f+ f. {4 Q$ ?8 B" Q* T, Q4 p : H7 S* Q( }+ {>> FundamentalUnit(M);3 A! j: `/ W, x& U
^9 Q3 i5 A& N2 V) o5 o8 ^' p" ?6 j; F
Runtime error in 'FundamentalUnit': Field must have positive discriminant * t$ i8 |$ S# p% J * \: K- `7 `4 j: E3 : i( Y' d1 V8 \: I ; Z1 G1 l0 D Z$ u" s- M& m>> Name(M, -3);" p2 I* H% h( `, o3 T
^! _: Y1 P l+ [; p/ ?/ u
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]$ {) V6 k: A1 k0 n
9 z+ h9 Y0 \! l: P11 P4 T7 }$ J0 ?
Abelian Group of order 15 T q1 N% g8 Y+ K3 ~+ k! Y
Mapping from: Abelian Group of order 1 to Set of ideals of M + M$ m) _8 ?/ AAbelian Group of order 1) S3 B' V% G2 G) p$ d; t& j& T
Mapping from: Abelian Group of order 1 to Set of ideals of M ! P9 J4 `" M- c2 X- O4 p Y1$ o j& ^+ |: P+ W( y2 U
1 * A0 @ k( g; i6 j; \Abelian Group of order 1 & g4 \( O) P. O7 W8 }% g1 J6 ?Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no" F7 S. \$ B+ H, q( T$ U. B! u( z
inverse]$ j# Y5 N* U; ~2 }5 @3 |
1: t9 _; o/ x W2 D# P
Abelian Group of order 18 K e! f0 h6 c) f5 v
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant3 ]4 D- F9 \* E( v' ^2 c
-3 given by a rule [no inverse] f+ r; V# z4 P) VAbelian Group of order 1# q% t4 K3 k. r4 u9 {7 W- z0 ?
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant # P/ ~! |6 h0 u-3 given by a rule [no inverse]5 e/ T' m( ^7 ]4 c
false 5 Z, k# S( T0 B' O+ mfalse
本帖最后由 lilianjie 于 2012-1-10 17:41 编辑 " `4 e: D2 N, L* @& L
lilianjie 发表于 2012-1-9 20:44 0 a& P/ Q% [, \, u分圆域：2 D' L8 {# ?4 p Z0 K0 n
C:=CyclotomicField(5);C; 6 ^( w) Z; {- H+ O4 D- QCyclotomicPolynomial(5);
+ q. b5 K* Z x9 E
5 d1 F! q" Y* g; w2 u分圆域：* B8 w& ~1 B* I2 l' y4 @9 T
分圆域：123 l. q7 U5 C# q$ ?% Y
) B% X9 h) ]3 c; ]7 o' f' ?) H q+ i
R.<x> = Q[]5 U- e! j" p9 Y& _
F8 = factor(x^8 - 1) " P A% L( Y8 A6 w' \F8 7 v u& K/ o% I2 \6 x. G6 U4 |0 L/ B! j; o7 o+ v
(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) 5 I! p' M9 F0 W$ J# {
5 \- A7 M& V6 h3 v
Q<x> := QuadraticField(8);Q;( a8 H" m) o2 n
C:=CyclotomicField(8);C;, \2 W* ?- m, c0 M# F+ `
FF:=CyclotomicPolynomial(8);FF;7 |3 t! k) q7 I6 s: J
, h7 [. w! K; A, \/ m: p1 lF := QuadraticField(8);, `2 W9 _% t( R ]- z
F; 3 z6 i: r: v: s |( ZD:=Factorization(FF) ;D; 9 u% L& L( `4 HQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field2 B. e3 T* U9 ]( I
Cyclotomic Field of order 8 and degree 4 3 N+ O; w! C) S. d, m$.1^4 + 1 : h, ]; W$ @% O n/ m& D9 fQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field $ J9 o% R, E& F4 {* N. N- k[- O& m+ X0 B- H N2 H# p
<$.1^4 + 1, 1>5 S" m! E! P! `5 W9 n- L
]) \7 d5 [( [ |$ x3 Q! G* O
9 g- Z" `/ @; L; J0 ?# ] e0 yR.<x> = QQ[] 1 z9 ], ?$ g! }% q2 ZF6 = factor(x^6 - 1) 1 O9 S% S! a5 a: i) Z. d1 `5 gF6* c% T `1 L: r5 j% n
# W; p E& ^6 ?: Y* `
(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) ; V* M1 X$ s( [- X. \- o0 e * Q: _) y0 Y) s4 _Q<x> := QuadraticField(6);Q; , S4 U& d. s6 Z9 `2 ]C:=CyclotomicField(6);C;- m$ I! M3 Q5 m7 |
FF:=CyclotomicPolynomial(6);FF; ! z5 L) r# [$ I P: D8 _' h% ]) M3 q, o3 p4 w: R/ b! Z4 M
F := QuadraticField(6);. E* \/ a$ m- K0 D
F; 4 V1 |+ c2 C" y( C6 iD:=Factorization(FF) ;D;% h: Y' K* Q [. f, I- X
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field : p7 D' X8 L, ] u4 i: u7 k' BCyclotomic Field of order 6 and degree 2 + O8 F e5 W2 C& m& ]* N2 V$.1^2 - $.1 + 1 # \$ E4 A* {% V: a1 x- PQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field4 f2 v9 k" J* X$ _6 T
[% M$ Z8 n$ G; O
<$.1^2 - $.1 + 1, 1> " x0 N; s1 H5 P3 Q] : L' j7 ?+ @6 S" w! {7 l: \$ u. v$ F/ I `* I) G B
R.<x> = QQ[]. |$ o7 s! p% n3 C: @
F5 = factor(x^10 - 1)) V2 T O6 Y- q d
F5: Z/ Q& V: }; I0 O9 R, w: n
(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + ( E2 r* G0 i; A4 x" n" W/ P1 e+ f1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)& H/ c0 f/ q, Q3 } x
3 ~) ~1 Z" R$ s/ v( `' pQ<x> := QuadraticField(10);Q;" X& M, [, N7 ~# u M
C:=CyclotomicField(10);C;9 c$ W/ A' W4 _+ B
FF:=CyclotomicPolynomial(10);FF; 8 ^" |8 T+ S4 V, g, g ( R" O Y, C4 N. Q, EF := QuadraticField(10); 3 Q3 | Y+ z3 ` h* w( m: [" ~F;" b6 a& e4 a$ x9 y& h" u) S
D:=Factorization(FF) ;D; 9 y% {5 F2 e, M4 kQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field- y& Y: g& o6 e& |% ~8 a# w5 ?
Cyclotomic Field of order 10 and degree 4' F" |' _- @( @. H
$.1^4 - $.1^3 + $.1^2 - $.1 + 1 ( k& B2 @. R% O- gQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field & k5 W5 \4 M8 E. d; ^' q- n[ + j* _8 Y9 \; ~ <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1> $ H3 @2 R& ^! |/ R2 X( \3 @]
IP卡
狗仔卡
发表于 2012-1-4 17:41
转播
淘帖
分享
收藏
支持
反对
微信
提升卡
置顶卡
沉默卡
喧嚣卡
变色卡
抢沙发
显身卡
