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虚二次域例两（-5/50）

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lilianjie        

43

主题

4

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204

积分

升级  52%

  • TA的每日心情
    开心
    2012-1-13 11:05

    • 签到天数: 15 天

    [LV.4]偶尔看看III
    跳转到指定楼层
    1#
    发表于 2012-1-4 17:41 |只看该作者 |倒序浏览
    |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 - q8 b% n2 o" g( g
    / ]4 d3 }) Q& N& Q9 s9 W
    Q5:=QuadraticField(-5) ;  U( \  T) M8 ^2 \& G0 S4 h$ ]
    Q5;
    ! b( e2 H" ~1 N* \/ J, D. r4 o
    * \) ]6 O4 J% f  Y6 Z. Y' |Q<w> :=PolynomialRing(Q5);Q;  M5 M& \# ?7 C" Z( z# E) o
    EquationOrder(Q5);
    6 O% x/ N/ v. w' J$ eM:=MaximalOrder(Q5) ;; B$ ?- `  E+ S1 V# w, v. |' E5 F
    M;
    . F" r5 V' {/ M! ]9 _* K0 L9 L: nNumberField(M);7 Y) B& O$ z3 e! ?
    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;
    , M+ b& j9 q+ Z- N- M1 Y1 s, DIsQuadratic(Q5);
      o. r4 H. O2 m3 W& Q/ r* ?6 iIsQuadratic(S1);
    , l8 k. a$ B0 P$ u: q( qIsQuadratic(S4);
    " |/ t( x/ U; _* VIsQuadratic(S25);
    1 G5 x$ u/ y" W3 ^IsQuadratic(S625888888);
    # b! l. c+ l, r* H, G4 u" N' HFactorization(w^2+5);  
    9 F. `* K% f, F& S- iDiscriminant(Q5) ;0 F, q, {- U, _( m
    FundamentalUnit(Q5) ;. }" ~8 A' }: D: V7 T+ r
    FundamentalUnit(M);* f# g$ b, K" S" K( j8 U( E7 r
    Conductor(Q5) ;0 b: w( X2 ?; w: J% R  P- N" Y, U
    3 w' V: a5 A/ A
    Name(M, -5);
    ' }+ ~7 f# b4 A7 n! a$ J6 D  oConductor(M);
    & b4 l( v& m1 X5 j+ E, u1 bClassGroup(Q5) ; $ X6 B8 e+ j- R0 p
    ClassGroup(M);7 }) M3 H8 e# |, J0 m0 y/ J  c" ^! s
    ClassNumber(Q5) ;5 I- Y4 w4 d$ c) x! Q  `
    ClassNumber(M) ;
    : O9 Q+ L/ \- p% k4 `: |9 qPicardGroup(M) ;4 h, N3 o& @- M' ?9 t% Y& T. y
    PicardNumber(M) ;9 U$ r' O5 O7 _5 n6 p! W1 I
    1 q3 w# e* }: |8 f1 b
    QuadraticClassGroupTwoPart(Q5);
    ; x1 S) ~0 C3 K3 l& `8 w5 WQuadraticClassGroupTwoPart(M);! g4 ^- ?- M: f4 ^- {
    NormEquation(Q5, -5) ;% g7 n, _- O  b8 F  _# H
    NormEquation(M, -5) ;9 e9 M: k* X/ n7 }' P% j( @1 I
    Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field+ b% r7 s7 @" R
    Univariate Polynomial Ring in w over Q5( E  e5 |- Q! Q& Y. u' s
    Equation Order of conductor 1 in Q5
    6 p+ A! H" a1 b- [% m) zMaximal Equation Order of Q5
    8 z! ^5 g. W$ yQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    5 B9 D- ~6 @6 S$ h9 M' u! yOrder of conductor 625888888 in Q5/ q# D! _! q4 E7 b, X: P
    true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field1 b$ {- {. _  p) h# ~! y; `
    true Maximal Equation Order of Q5
    ( ?0 z- M, O7 S6 S1 wtrue Order of conductor 1 in Q5. t' E  b- d: n# k* `9 s  S
    true Order of conductor 1 in Q5- w4 b+ b/ ?( u+ f9 E# f2 f% d3 h
    true Order of conductor 1 in Q5
    . h8 d( @& s9 x/ \. u' |# U[* t7 T2 _% w7 v/ s
        <w - Q5.1, 1>,
    # P: y" `, e% x9 ^    <w + Q5.1, 1>; N2 P. }& G/ l. {3 ^$ J$ B4 p, E
    ]) M$ B* f5 F: e  C: j5 m# B1 [
    -20, x# y; W0 B. {  T
    3 _9 W5 o0 S+ S5 f  X% j
    >> FundamentalUnit(Q5) ;
    . F8 C3 W& v2 A( G. ?                  ^2 H; Y7 [, d* x# ?* ?+ Y& {) F
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    ; p3 {) ?  d: I* d
    2 x- T/ N" a( t+ Q9 y5 `2 |2 p# E6 @( r* u) T6 l2 Z! W& K
    >> FundamentalUnit(M);
    : Z( R) O& a, L* L' G                  ^+ H8 C5 T9 L% g: e( b0 I  X; Y
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    & ^- i6 H6 i* H4 n) K0 l: |; D. G1 a8 A) E  c9 n7 y& R
    20; U6 O6 q$ W3 G/ ~/ C

    ! L8 {5 Z, o0 o* F3 Y4 Z& D9 X>> Name(M, -5);
    # o* n8 J; _4 G4 J       ^6 _2 b* i: P# E& Y/ j6 J5 m
    Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]% [. P! j$ y; k- R# T$ g1 P' D
    : j. ^7 Z1 J6 X
    13 Z$ y" _# ~. d, N5 \
    Abelian Group isomorphic to Z/2
    0 q$ ^* N8 [$ w  Z- ZDefined on 1 generator& J! y4 k& U0 W# @
    Relations:
    3 d0 i6 M7 P5 K. Z  K    2*$.1 = 08 g) R! U: A7 @7 @# y' j, C4 Z
    Mapping from: Abelian Group isomorphic to Z/2
    . [! i! W  R2 |* `! o3 C4 {Defined on 1 generator: t2 \/ B2 [+ g" A( @0 {* o
    Relations:) ~8 O6 f2 s9 ?* p  g( _9 p
        2*$.1 = 0 to Set of ideals of M
    + p) j4 l: @- u3 a0 L: H% ]Abelian Group isomorphic to Z/2
    # S6 Z% X% S# j, o$ [* ~; wDefined on 1 generator& a9 Y: |# S2 K9 w1 m/ n0 Y
    Relations:! g& w9 ^; X$ J" r8 q3 b
        2*$.1 = 0  [: @! l. a) O! `& |
    Mapping from: Abelian Group isomorphic to Z/2
      _) \. m6 z6 r2 `6 a4 G" oDefined on 1 generator& b5 @5 a3 C/ f3 ?% J3 D+ Z* e2 H
    Relations:
    % V8 i! w# }7 u: \$ t( w    2*$.1 = 0 to Set of ideals of M/ S+ d2 w7 |; [$ z8 Z; r; [0 m9 W, y
    2
    - o, l( `$ h; z! m  K2* F# G' \/ Y8 i7 ]1 u
    Abelian Group isomorphic to Z/2" b" d3 T# c, Z- ?, |9 q8 }
    Defined on 1 generator; ^) R% g% K$ F3 G$ a
    Relations:
    ( g' |5 D: y. L: A    2*$.1 = 0' I; g+ u* Z( \2 j
    Mapping from: Abelian Group isomorphic to Z/2
    ; f% R8 ]! ~0 I; b( ODefined on 1 generator- a3 O7 o5 f& z. t
    Relations:. F7 B8 i! M9 l0 M8 T' A/ y
        2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]$ Z6 j& G% F* ^' Z' z
    2
    " K) M, l2 B6 p, h( TAbelian Group isomorphic to Z/25 F2 C# @" l! S
    Defined on 1 generator) |" Y1 A' ]; j6 L' m/ d) V
    Relations:- g) Y  V' V0 F5 e5 h
        2*$.1 = 0: c5 I/ [+ E5 b- `0 p5 F
    Mapping from: Abelian Group isomorphic to Z/2) I' S" m  X  x: t9 M. }  O  g
    Defined on 1 generator
    , u  c& _& L. J' @- KRelations:
    " ~$ t. s! C1 C$ L! z# j    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
    * `$ I1 O: y4 Q+ i* Vinverse]
    , ~, E. k6 c) i$ p  z6 UAbelian Group isomorphic to Z/2- }( x8 p- ^6 L3 m0 W/ }
    Defined on 1 generator
    9 D' A# |5 I' _, r" URelations:1 t- ], t/ |$ w! A
        2*$.1 = 0% v' }9 D. }& q: c4 c5 j& Y
    Mapping from: Abelian Group isomorphic to Z/21 |' D" c* T0 c) g
    Defined on 1 generator
      S  X) M+ ~  q* K/ cRelations:1 ]" @& g0 K% P0 o( v( V5 N/ R( w2 N
        2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no   t7 Y  V) `& c, }
    inverse]
    $ @" X% T0 R& t$ O( Y1 C4 T  ~2 \0 ]false
    % g4 i6 X) x0 p7 z9 L/ O# H% m3 Z8 c' Qfalse* `3 @% w4 L' C5 F! p6 B
    ==============: o& e# |; h( q( ~# k

    : _2 J2 p2 ^$ J
    & q$ r6 N7 z3 W9 ^5 C8 JQ5:=QuadraticField(-50) ;
    ( [- J9 \5 g& o! s( v- b$ l6 sQ5;) l4 q; _: n8 e8 F- b

    % @1 e( K% @7 g$ [0 W" _( w2 C8 f" xQ<w> :=PolynomialRing(Q5);Q;: `7 F; s. g1 w" M- d
    EquationOrder(Q5);; R: N' U' S% B5 z" F
    M:=MaximalOrder(Q5) ;
    ( _) L1 x* {5 I2 X. ~4 o$ @M;
    # K7 H/ b0 a/ JNumberField(M);, v; y' X1 M1 {7 D9 E4 ?
    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;1 d& ]+ a7 g6 A
    IsQuadratic(Q5);
    . B! j2 w( Q( e; ?4 a) j. l4 o. ?IsQuadratic(S1);3 s" H) N* R7 i: O+ F9 N* V7 D6 }
    IsQuadratic(S4);: l/ I) K# e2 N5 y9 U, u  u
    IsQuadratic(S25);# }$ W) t: \) N' [
    IsQuadratic(S625888888);0 }) Z+ G. R+ w+ H7 l) }
    Factorization(w^2+50);  4 d- y4 \- k6 Y2 L
    Discriminant(Q5) ;( |/ t% j/ w3 ], Q# |! Z9 i
    FundamentalUnit(Q5) ;0 H. q* s$ \, `: l0 s5 M+ i2 t
    FundamentalUnit(M);5 G1 t- |6 |, e! O
    Conductor(Q5) ;
    , k! e( w6 s) f+ @6 w! O1 q4 `! w2 d, g7 o8 E& o# D. n
    Name(M, -50);3 B( z% j! N# S  H- I9 j4 l& [
    Conductor(M);
    3 j+ b9 w, D6 vClassGroup(Q5) ; / j0 {7 U, i7 B1 o+ k- F
    ClassGroup(M);
    - Y# i7 D! w6 A" }0 s5 g! p# _ClassNumber(Q5) ;
    ' S9 A3 d8 t% l/ o" j+ N3 q* LClassNumber(M) ;
    0 y1 \/ j2 ~. w* XPicardGroup(M) ;
    " j8 I! B. x; ~( L0 Y6 SPicardNumber(M) ;
    5 h* J" B. a. u, k) I! q! A
    $ ~5 _7 b) `& R6 ^QuadraticClassGroupTwoPart(Q5);
    " r9 `1 J  `! z( w4 Q* XQuadraticClassGroupTwoPart(M);
    ( \+ X- y$ y; z! K/ c6 y" Q; pNormEquation(Q5, -50) ;( Y; z* Z- [! s8 o5 r
    NormEquation(M, -50) ;$ j: s# z* m6 u4 N0 U

    ' L% ^% t0 H! Y# [. f/ bQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field; D- G0 r' D0 U* G$ {) E# J: |6 \
    Univariate Polynomial Ring in w over Q5
    1 [4 n- C7 H! a& r9 a0 Q$ Y: E; O7 XEquation Order of conductor 1 in Q5
    * x; r5 ~/ ^5 QMaximal Equation Order of Q5
    - W# X* `% k, A/ T/ S; dQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field. I6 }) K8 S- i7 q; @0 n! i
    Order of conductor 625888888 in Q5
    ! h' z' P2 }* Ntrue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    0 H, T6 p$ J4 Ptrue Maximal Equation Order of Q5
    ( K0 h- j. w' Z# ltrue Order of conductor 1 in Q5, F7 `* i2 B! K9 R6 x! n
    true Order of conductor 1 in Q5
    ! ]; G, r9 }+ s! [! Ztrue Order of conductor 1 in Q53 w2 \$ ~+ B1 X0 v. r. \) [
    [
    4 C4 \2 i$ }7 l/ F- W    <w - 5*Q5.1, 1>,6 r( ^1 f2 n: D* C- r7 ^
        <w + 5*Q5.1, 1>: m' a) n' g# r. I# }
    ]- u: }+ L  {3 R6 f: G" W1 b, K
    -8
      ?& J+ v, K& l/ V" K5 m$ T7 }4 w# }% H
    >> FundamentalUnit(Q5) ;2 a1 v! Q6 H* i) X2 T- x0 n9 V
                      ^8 ?7 J  j( r- e. T& \  \
    Runtime error in 'FundamentalUnit': Field must have positive discriminant( k4 U; ?; L% t6 t1 U
    . n$ m& _/ G! p9 B$ K% Z& k# w2 Y" D

    * A# I0 C1 z: N" V; M- v" M2 P>> FundamentalUnit(M);
    ) ]. c- N0 K. L4 p                  ^
    0 x& A' h; @  ~) s$ i' HRuntime error in 'FundamentalUnit': Field must have positive discriminant
    % I' F& w0 ?, S; g+ g, t* m9 P1 f9 G" ]  V  }( X% X( h
    8
    : h: i3 b% t" h9 J& p3 t/ k9 J; k' c' l- r. c9 c
    >> Name(M, -50);2 z) }/ C# R$ r  u5 U
           ^5 O. }6 A1 _6 a- E$ g+ N& E
    Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
    # A+ I$ a, [" l4 ~8 f9 h0 y3 K, I9 ~. N" P$ |" }
    1
    & L% X/ P, \$ ]; L/ w; x) EAbelian Group of order 16 N( k4 i% i' `' h! ?' d0 s8 }
    Mapping from: Abelian Group of order 1 to Set of ideals of M9 r9 T+ k7 v) G7 C# b
    Abelian Group of order 1& J. l/ o* E7 X2 I" g
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    & f/ w& W) t' t7 k! _19 p& e1 |0 \$ E7 _) K/ p
    1
    # d0 B: \7 d, p% r; @7 [; a+ q) pAbelian Group of order 1
    3 `& N! n& S4 o  [% i8 \( JMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no6 a8 u1 ]  j- j9 y
    inverse]+ P) W- {, ~* K; v$ v$ U2 j5 {
    19 }, y: x5 _. t* f" I' @4 V! ?
    Abelian Group of order 1& n& y  X, L4 A* K6 Y
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant/ H& w9 t6 D/ M3 W! ]# c) T7 X- V
    -8 given by a rule [no inverse]6 c* i  ]" U5 z, v* O% a* ?/ f
    Abelian Group of order 1
    # F  w6 S& M9 F& MMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant& @) T# _: c1 R' ]) q* k
    -8 given by a rule [no inverse]
    ) `% r4 _0 p& o  \# |false
    9 V. Y- }4 U# N# R. ~false
      f8 @& n& L! _9 i
    zan
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    lilianjie        

    43

    主题

    4

    听众

    204

    积分

    升级  52%

  • TA的每日心情
    开心
    2012-1-13 11:05

    • 签到天数: 15 天

    [LV.4]偶尔看看III
    2#
    发表于 2012-1-4 17:51 |只看该作者 |招呼Ta 关注Ta
    看看-1.-3的两种：
    5 X8 }, z# _# E% @) ^3 j' x% E$ W( z4 A1 i; J: U6 n6 z6 O  s- `
    Q5:=QuadraticField(-1) ;' }9 H4 A! |0 h% E
    Q5;
    6 N! u( Z3 Q+ l7 b0 c0 f& V% _% @: q3 Q: W# g" o1 v
    Q<w> :=PolynomialRing(Q5);Q;
    % K- X' k& g/ r- `+ \; B7 W6 AEquationOrder(Q5);
    + u3 o* N& d7 ]7 @" ]3 F+ QM:=MaximalOrder(Q5) ;/ a7 ^7 F2 G/ }! v" G+ j, B2 i; |) ^- P
    M;4 [$ g7 F; I6 g7 j2 ~: _1 v0 p2 O& ?
    NumberField(M);, X$ Y; m; @, G: }; a
    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;
    7 S  E6 P7 B2 ?0 \# o! [4 X% ~IsQuadratic(Q5);& m! e& S3 ^; U! g) Z# V
    IsQuadratic(S1);8 \1 f+ {4 w( d: u. t! j8 [
    IsQuadratic(S4);
    - |8 P2 U" I: b4 _) k9 a. ^3 GIsQuadratic(S25);# J+ z; ~9 l2 @
    IsQuadratic(S625888888);. Q) o0 ~* a1 F8 h- P" b- b
    Factorization(w^2+1);  
    : ?- J/ p- E" x/ m9 o5 R' K. XDiscriminant(Q5) ;) K5 o+ ]6 Z% _- j4 l
    FundamentalUnit(Q5) ;
    8 [8 Y, c# Q6 }, ~. r4 TFundamentalUnit(M);
    % i) u8 E( F! RConductor(Q5) ;6 J0 g" \  h, `+ _  U4 \7 `
    ' h9 W: L" q0 @  X& W
    Name(M, -1);+ _' ~2 _' R+ ?' S2 B9 e
    Conductor(M);
    % @' X3 w, C2 oClassGroup(Q5) ;
    2 M% W1 F* H% mClassGroup(M);
    / i" G4 Q  N7 }& FClassNumber(Q5) ;
    8 I5 G" p+ W1 k$ K7 ?# ~ClassNumber(M) ;' x% j: |8 |! F0 i4 v, s
    PicardGroup(M) ;: ^' ^- C! ~# d% y( O* t
    PicardNumber(M) ;
    9 O6 ?! X" U  H4 |" @0 d/ s# e1 ?8 T0 H
    QuadraticClassGroupTwoPart(Q5);
    3 q5 W. K8 h% V' a$ d9 x9 MQuadraticClassGroupTwoPart(M);9 H0 {9 ~0 b. `/ u2 u. I+ H
    NormEquation(Q5, -1) ;
    4 Z( T5 [! A: y" JNormEquation(M, -1) ;
    8 D9 ^8 G# J2 s) d- z3 q& X( y
    ! b  V6 G7 o/ v9 sQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    ! I7 a7 k# \; o) _$ k  n# [Univariate Polynomial Ring in w over Q51 ]1 O' y0 |! l, v
    Equation Order of conductor 1 in Q5, ^2 u9 r3 P4 w" H1 X! `
    Maximal Equation Order of Q5
    ( J3 B) i: B; m  H0 R. z! }6 RQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field( p$ H+ F( \6 w. P4 f
    Order of conductor 625888888 in Q5
    / u# n1 w" M. o# \# t2 E% w. ptrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field7 @' m1 F  r: O" R! p
    true Maximal Equation Order of Q51 `4 v$ g: F0 t9 y  F8 k: V% u
    true Order of conductor 1 in Q54 {) h8 R  c$ W# p/ R1 E/ Z
    true Order of conductor 1 in Q5
    2 [8 z$ k, p1 ?& Y: Q6 G$ Utrue Order of conductor 1 in Q5+ j- e/ q9 X: K& ?
    [( g/ Q& M( F* Z( L( N  l
        <w - Q5.1, 1>,
    + ]# _/ w) v. W" h! I& e7 R$ T* ^) T) t    <w + Q5.1, 1>
    9 J1 q1 I4 L  o" ]  b]' ]0 ~# o0 I* M. ^
    -4
    3 Z8 l& [2 t, ~1 E
    # m" X4 }% d7 J' l7 r>> FundamentalUnit(Q5) ;
    2 t" V+ c; h7 Z: L                  ^
    ' ^" W- [. Y; U- p; p, l. ?( KRuntime error in 'FundamentalUnit': Field must have positive discriminant
    3 P" ^2 Z1 }. n) M! [+ e5 n- X2 ]" i1 {+ B4 N! {3 b2 N2 X

    . f( F; b! Q9 N4 ]2 d& u>> FundamentalUnit(M);9 L% w& @- I* l
                      ^& h( F. U% l9 i, M6 l
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    ' ?) F5 J4 j' E9 V
    2 |) y; ]% X+ P4
    " d! F5 \8 a6 [: @; K8 \0 j5 n* k* R5 V+ z! p
    >> Name(M, -1);
    ! {5 W  E( @% o) T2 p+ u9 P; ]0 P       ^/ J" V& i# B4 w& y  m! \% ?5 `
    Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]; N+ Z" h+ x# a- p# n0 s

    8 u- a; u# j2 D; h7 a1
    9 R( f. I2 d& s- G" nAbelian Group of order 1; X, ~8 ]$ {4 e: d; @
    Mapping from: Abelian Group of order 1 to Set of ideals of M1 P3 k. R! a0 J
    Abelian Group of order 1
    , ?: I1 u5 m3 ^  _( w2 `Mapping from: Abelian Group of order 1 to Set of ideals of M6 c: Z1 y9 u3 H; J6 x
    12 v' x' l6 Y+ v7 A1 E6 `( G
    1. r9 X% C+ y" z/ H
    Abelian Group of order 1. q; a4 L# ^1 B: q) C6 A6 ~0 f/ G
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    & a9 h' ]1 \1 k5 uinverse]
    8 b/ M$ Q! x% |4 w. ^+ Y1& \% Z3 m$ o$ P) t+ E/ e
    Abelian Group of order 1
    5 W3 l' n) r- @' J* u3 aMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant  [/ o7 P* I* A5 _
    -4 given by a rule [no inverse]* u! ^+ J) x" L! C9 `
    Abelian Group of order 1% I" \) j* r, `8 e
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant* ~4 u$ |, |, z$ j/ D) u
    -4 given by a rule [no inverse]* F  T8 m: Q$ Q) x. `+ b, Q
    false
    4 B. e) z2 {; b# ^5 rfalse3 ]2 N; ~0 H, z" K  z: M
    ===============
      A: A6 I, j3 Z  k' l9 g, S
    ' O2 z, K% @5 {: e4 tQ5:=QuadraticField(-3) ;
    ' @% k- [5 c; x/ t# Z9 b# I  n6 oQ5;  X+ k8 P7 a6 o

    / e# a! H9 \1 D6 E* f  [4 TQ<w> :=PolynomialRing(Q5);Q;
    ! u8 |  b. p' E6 \7 S6 F; gEquationOrder(Q5);/ X! S7 I+ Q% w1 j+ z$ v2 S7 e
    M:=MaximalOrder(Q5) ;
    7 Q" O; T( k8 C# Y4 H# ?; r, vM;5 Q6 K) F* d/ O, k
    NumberField(M);  V. S- s. V' [# z  K) p8 N& a
    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;
    : `  B; A; K0 Z& k3 O, _1 _IsQuadratic(Q5);
    ) Q7 t; l" m5 X& vIsQuadratic(S1);
    , R$ \) h! F& v8 c6 `6 [8 L  bIsQuadratic(S4);
    9 G8 g; Z" ]) o. m) @$ gIsQuadratic(S25);& F6 ?" l8 b5 P+ w
    IsQuadratic(S625888888);4 L+ E, {  I1 J" h
    Factorization(w^2+3);  5 b' f1 C/ q, C0 I' S4 i% q# x
    Discriminant(Q5) ;
    " y% z$ A. @* O; i/ }7 x$ kFundamentalUnit(Q5) ;
    2 V8 a1 V0 ?4 P0 hFundamentalUnit(M);
    5 c$ I! Z' h  q+ e1 G, g) a& [Conductor(Q5) ;
    + e) g8 ~$ ^, W% V4 P+ @7 J1 a. K, t+ C
    Name(M, -3);, G& g/ B& ^& i
    Conductor(M);; k: R! g8 [1 h0 |! X: C  c+ I( s8 W
    ClassGroup(Q5) ; ' Y1 ]8 ?& n+ c& d, z- e
    ClassGroup(M);
    - d7 i# A9 b5 E  b7 ]ClassNumber(Q5) ;0 |$ o7 h& m1 H7 e
    ClassNumber(M) ;
    # Z3 V1 t7 d  m9 _PicardGroup(M) ;* l/ e: U2 ]# [. x8 ^4 c8 x( V
    PicardNumber(M) ;1 K* X4 A0 z! u  j- R- L# O
    0 A5 @. E, M" y0 ]- r( A) s! T  V& `5 M0 t
    QuadraticClassGroupTwoPart(Q5);, B* p$ N- I; M4 d( S- b
    QuadraticClassGroupTwoPart(M);
    5 U% `( G6 X5 b( J9 B  zNormEquation(Q5, -3) ;# M- N( R  e0 Y( M3 w5 i( L) l
    NormEquation(M, -3) ;
    ! G; X- E3 J8 c' l; ~, ]  B" p/ j$ |9 ^  K9 |# U
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
    ( R; g: [; R- i2 U; VUnivariate Polynomial Ring in w over Q5% v/ t0 U: N- t% q
    Equation Order of conductor 2 in Q53 u% w5 F* p* Q' K
    Maximal Order of Q5# A0 f3 w+ Y9 p' I! _
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field  T3 ~% M: o3 K' \4 Q
    Order of conductor 625888888 in Q56 ^, T+ Y+ T; R/ X$ U
    true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field: D7 l3 ^5 R5 o  [" f( C
    true Maximal Order of Q5
    5 v5 C; ?* I, @+ f8 Ytrue Order of conductor 16 in Q56 W' J% i$ W6 P; z8 b  q) x3 J
    true Order of conductor 625 in Q5# g) j$ ?  R+ j. f) f
    true Order of conductor 391736900121876544 in Q5  {' T7 U  t! F/ v2 s& B0 ~: O
    [
    ( o/ T$ ]' }+ N- `( }7 Y, ]    <w - Q5.1, 1>,
    $ Z5 ^5 R9 z3 `, L1 Y5 r2 Z6 o" H    <w + Q5.1, 1>
    & r& i( v1 P& G; Y7 y! n: B]
      C  O9 j% h& O" Y1 ?-3  n# A$ E; i( [
    - x4 L, m# l9 z. ]. w$ z
    >> FundamentalUnit(Q5) ;
    & ]0 z2 X' D6 q: ^                  ^
    " u$ D. X; `$ BRuntime error in 'FundamentalUnit': Field must have positive discriminant2 \( A* A- e, |; [: l; n
    4 v8 j) Q8 O9 k# c
    8 j+ U+ v1 x- |4 g# n
    >> FundamentalUnit(M);6 R# Y, O* v+ L5 w9 e
                      ^
    3 l& q: _' ^+ u% xRuntime error in 'FundamentalUnit': Field must have positive discriminant0 D- @4 \: h. u% J4 o
    + P- d) M- @9 V# A* v/ i6 v) y9 q
    3
    , Z0 y" A. O/ b- F1 @* a0 b( m
    9 k) |* {  |2 l  W+ y' A2 {$ r>> Name(M, -3);
    7 o* q9 B, k  e+ T$ g1 {, `       ^. s8 R  c5 y) I( G
    Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]( P+ N6 G# E/ M" [
    4 k% N! |. ~' b& ~
    1
    & k  S1 x& a) D, B5 m6 r4 JAbelian Group of order 1
    ( n5 ]+ G3 V% w4 V& s3 ]" v8 VMapping from: Abelian Group of order 1 to Set of ideals of M4 E9 G8 v9 N% {* Z' _* `$ S- r
    Abelian Group of order 1; ~# ], a( Z+ Y8 O
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    : V8 F2 j( B) o1 x* j4 r1% C- T- k! D4 i1 B, T, U
    1  B) I: h1 }; b; t- k" ]7 `8 B3 R
    Abelian Group of order 13 {4 s  f$ v! n( N
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no/ w2 |- [, _0 x: [! H) t
    inverse]
    8 ~( w) Q% k) k5 b# C1" I: }8 @( W$ D  a, x
    Abelian Group of order 11 \( S2 b/ q; {* Z4 C& j. `
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant1 O7 _( r. ^. D& o! Q
    -3 given by a rule [no inverse]6 Y3 O3 a) X2 ]' a7 G1 R  f8 {& e
    Abelian Group of order 1; |) q' G$ M! s( m9 l( ~4 u9 y
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant8 F) _, L+ q" H  L% B  |) `7 Y( n
    -3 given by a rule [no inverse]
    + ~$ _! R% K* A; J  lfalse
    2 y8 {0 s! S$ Z* r# J4 nfalse
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    孤寂冷逍遥 实名认证    中国数模人才认证   

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    2015-9-4 00:52

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    [LV.9]以坛为家II

    社区QQ达人 邮箱绑定达人 发帖功臣 最具活力勋章

    群组: 数学建摸协会

    群组: Matlab讨论组

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    群组: 数学建模

    群组: LINGO
    3#
    发表于 2012-1-5 08:36 |只看该作者 |招呼Ta 关注Ta |邮箱已经成功绑定
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    [LV.4]偶尔看看III
    4#
    发表于 2012-1-5 13:02 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-5 15:20 编辑 3 v' m8 N) b+ r/ R0 n
    & f. h/ D; Q6 M0 \5 {$ Y' b
    Dirichlet character
    ) U* |/ k* S9 N7 lDirichlet class number formula
    ) v7 `6 ~- x4 f0 E8 W4 ]2 `5 X- @3 s  V# x8 t
    虚二次域复点1，实点为0，实二次域复点0，实点为2，实二次域只有两单位根, a- j# g9 m( `3 N9 Q3 j$ _
    * y) p$ M4 ^7 x1 w& `" C# O6 b
    -1时，4个单位根1，-1, i, -i,w=4,            N=4，互素（1，3），    (Z/4Z)*------->C*      χ（1mod4）=1，χ（3mod4）=-1，h=4/（2*4）*Σ[1*1+（3*（-1）]=12 H' X5 N7 |, j1 V0 Q9 F* N* N
    4 l/ D/ H& X$ x( u4 o) s/ K% A" b
    -3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，2 K% G2 H/ x+ c; t
    h=-6/（2*3）*Σ[1*1+（2*（-1）]=11 M: A+ B/ G0 `" J

    6 z& f% e3 [( r7 l! C" Y-5时  2个单位根                              N=20   N=3  互素（1，3，7,9,11,13,17,19），    (Z/5Z)*------->C*         χ（1mod20）=1，χ（3mod20）=-1， χ（7mod20）=1， χ（9mod20）=1，χ（11mod20）=1，χ（13mod20）=1，χ（17mod20）=1，χ（19mod20）=1，6 S; X9 d7 @& f$ J" l5 Z; h- x  Y/ D
    0 }8 {: r8 B) K* v$ r
    1 z$ @0 U" h5 [4 ?" q6 B- G) {; q1 L" o
    # W+ a$ b# Q3 K( S# _
    h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=22 f/ G. R. a+ e7 C4 a8 D  H+ U) a

    9 g( }) Y" h1 c7 i/ }/ `" m. ^" c# y; l6 d$ u3 B
    % O  |: j( C  o$ ]
    -50时  个单位根                          N=200- L7 R7 ^, i0 n2 q/ Q; ~+ N
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    2012-1-13 11:49

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    [LV.3]偶尔看看II
    5#
    发表于 2012-1-5 13:51 |只看该作者 |招呼Ta 关注Ta
    Dirichlet character

    21.JPG (79.18 KB, 下载次数: 179)

    21.JPG

    11.JPG (74.76 KB, 下载次数: 178)

    11.JPG

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    [LV.3]偶尔看看II
    6#
    发表于 2012-1-5 20:37 |只看该作者 |招呼Ta 关注Ta
    pell   equation

    11.GIF (22.03 KB, 下载次数: 177)

    11.GIF

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    [LV.4]偶尔看看III
    7#
    发表于 2012-1-9 20:28 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-9 20:30 编辑 & l- Y8 v7 p0 y. y  B0 i( j7 m/ b
    , l4 D4 @( m: H1 d
    F := QuadraticField(NextPrime(5));. w) j+ L) ?$ P6 m
    9 A% H* F6 i1 @
    KK := QuadraticField(7);KK;
    + }" C, n, p. d4 h9 A$ S) mK:=MaximalOrder(KK);
    4 G3 G7 }- o& H8 L7 f! {# W! qConductor(KK);- N. l+ z$ J8 K7 f0 _4 n: k
    ClassGroup(KK) ;
      C- ]0 @3 |1 C# j0 d7 p0 V5 fQuadraticClassGroupTwoPart(KK) ;
    + ]0 X# |9 ~1 e# |8 SNormEquation(F, 7);
    5 V8 D  J: S7 x# C# R2 E" t, p/ _2 mA:=K!7;A;2 ]. _9 ]4 F6 S
    B:=K!14;B;% G* Z- a( L5 t
    Discriminant(KK)
    / B0 f4 p' j: l* X( ?& j/ ^+ ^2 N+ ~. k) J
    Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
    8 h/ A* B) {" N, m28
    / m: L4 p4 ?9 ^; a+ ~Abelian Group of order 1
    ! ~- c2 p. O5 E$ pMapping from: Abelian Group of order 1 to Set of ideals of K* V& B, m. d3 c6 k$ ]2 I
    Abelian Group isomorphic to Z/22 y% G( @  u# `- J
    Defined on 1 generator# D3 Y: V9 \4 T1 ]! z2 W
    Relations:
      I+ s: x6 i7 d" x( T) Y" }    2*$.1 = 04 H1 X) E4 n* D! v: n
    Mapping from: Abelian Group isomorphic to Z/26 I2 |# U+ a3 Q) Z. G
    Defined on 1 generator1 }8 i( V8 ?% |0 ]
    Relations:
    $ ~: t  o$ N0 T9 M* p    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
    ' S2 R7 t& ?$ i: n2 n, ginverse]5 q# H+ K" ]; ]- Z- T6 y6 L3 _
    false
    ) a: }( Q4 `+ m0 C& T/ t7
    / ^6 _9 ^2 C+ z. G* `# p148 v- h+ E8 H$ d  R2 O: E# t  I
    28
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    [LV.4]偶尔看看III
    8#
    发表于 2012-1-9 20:44 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-10 11:23 编辑 ! m3 s- |3 g9 v6 \
    ( U; g( t2 d! k# z# i: i9 E6 \% G
    11.JPG
    % z) E% Y+ k# ~; w' o& }" D$ T; N3 R8 {5 ]
    3212.JPG 4 O: W  i; v( D& U  x
    # ^- H# w' {) G2 X" S
    123.JPG
    ' x3 n! t" V4 p2 r5 V# W
    % Q1 ~. p" H% j' R9 H9 X- E分圆域：
    / L4 n  J; N9 N! T& i& mC:=CyclotomicField(5);C;
    ' {+ m& e3 z' HCyclotomicPolynomial(5);
      x* P1 I8 y6 P) n& MC:=CyclotomicField(6);C;$ V1 r6 ~7 }) j
    CyclotomicPolynomial(6);! `1 o( Q5 _3 A; F  g
    CC:=CyclotomicField(7);CC;
    0 J+ T9 P, R% \( DCyclotomicPolynomial(7);
    & t% W& t% h& U2 i) R; A( V  IMinimalField(CC!7) ;/ J0 P3 @# H- M' {: D
    MinimalField(CC!8) ;: Y, F' ?8 \2 \8 \
    MinimalField(CC!9) ;
    2 b" Z6 S, V& _# gMinimalCyclotomicField(CC!7) ;9 n" u" j) d- l& F  H
    RootOfUnity(11);RootOfUnity(111);* Z& E! e0 S# E
    Minimise(CC!123);
    ' z1 K1 M" x8 A- {4 H. m1 D* fConductor(CC) ;2 E& h1 _5 A8 S2 z9 R- x/ G  d
    CyclotomicOrder(CC) ;. ]! t! h/ M8 T% S2 n' |
    5 K" r1 l" ?+ U  ]- {
    CyclotomicAutomorphismGroup(CC) ;- U1 n( I3 R0 s7 H: V
    ( h4 f) `% f8 p& A6 c8 \
    Cyclotomic Field of order 5 and degree 40 o8 r/ D/ m! W0 a, {0 }6 C
    $.1^4 + $.1^3 + $.1^2 + $.1 + 1/ I1 M0 u, ]7 C) J0 E5 U! r& M
    Cyclotomic Field of order 6 and degree 2/ z; X9 e- u& s# g6 t3 X+ |
    $.1^2 - $.1 + 1
    ) i$ _6 x+ c% K$ r" }7 a+ V! lCyclotomic Field of order 7 and degree 68 \5 R/ h" w2 w2 Y5 y
    $.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
    $ s; g! c% m: G( f9 JRational Field
    * K3 ~" u9 k5 ^3 g/ _Rational Field
    ) z2 a- d! a' `* G) wRational Field# r0 U) u: Z) _; Y, C! b
    Rational Field
      l2 }$ s+ a' ozeta_11  L+ x; F8 p, Q! h- P: H. M: x  j
    zeta_111. L9 X4 S+ |- _/ a6 B8 D
    1237 Z* d& G* w0 [0 ]; r
    7
    : _- Y, `" [5 M( B7$ E# Z7 ?, ~+ U9 v! D
    Permutation group acting on a set of cardinality 62 x5 f% T# ?- J# y. ^0 f" \
    Order = 6 = 2 * 3
    1 f7 F% E% W, R    (1, 2)(3, 5)(4, 6); Z' i. I5 S6 q" ]4 _9 H* t0 O
        (1, 3, 6, 2, 5, 4)
    , c. H- A9 a# w2 N8 h7 dMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
    % Q8 ^) m9 |6 d4 r( XCC' z8 m4 ?! v& R
    Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $, / ~/ m! E  s4 r/ y" d
    Degree 6, Order 2 * 3 and2 C7 {" s; I4 z( \: ]0 p
    Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
    3 s$ D0 U9 L  {* g- X/ vCC
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    [LV.4]偶尔看看III
    9#
    发表于 2012-1-10 11:34 |只看该作者 |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-10 17:41 编辑 $ x0 j5 @5 j! r# u* W
    lilianjie 发表于 2012-1-9 20:44 & O8 S  s( o" w; b
    分圆域：* ^% _; @! D' [! u
    C:=CyclotomicField(5);C;) \# U$ x8 \, }
    CyclotomicPolynomial(5);

    3 Y# W! O6 e( Z5 }, {- K5 @2 @% h; S" ]
    分圆域：2 U) x) K7 ~# b. }
    分圆域：123. ]1 H7 h9 n) {% k
    # C# k0 w/ H1 j) S* g6 o
    R.<x> = Q[]
    / _" i, s+ i% t, H; GF8 = factor(x^8 - 1)( B* X/ j4 E  l* o/ Q- W* O) t
    F8; j7 Q$ v. J* |* h. f$ Q9 q, M

    9 y! Z: ?4 ~) G7 x0 F( `6 J(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
    ( M/ l  w! u' o6 U0 x! w7 ^* T+ L% {2 L8 z  y+ E/ L5 {) D* g! f
    Q<x> := QuadraticField(8);Q;
    ( c' R9 ]: d. gC:=CyclotomicField(8);C;- y0 P" ^7 O& x/ S; n
    FF:=CyclotomicPolynomial(8);FF;
    : B. A3 y/ Y+ I( T8 W8 v. u2 b  B% \
    F := QuadraticField(8);: r4 {) G+ @8 b& g) _' a3 K
    F;
    9 D) V7 Y8 @0 l4 l' n- ^D:=Factorization(FF) ;D;
    : i- ]0 o/ ^% Z( fQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    " e% |/ [3 c8 @# _2 a9 l' J7 n$ V* OCyclotomic Field of order 8 and degree 4+ {( r; B2 y) ?5 @" I6 m: N
    $.1^4 + 12 `' }: b; V( Y9 t" t6 t7 _; Q
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field1 X& ?* |5 D2 O, r4 m( N
    [+ z+ [, D1 ~; ?/ y; E0 h% T4 u
        <$.1^4 + 1, 1>0 }& W* [/ i# a8 ~0 P( Z
    ]
    ( c2 ^7 s+ m$ W, V' a
    6 r# R6 ~& L- w1 {R.<x> = QQ[]
    ( d; @7 q6 ^; _! ]F6 = factor(x^6 - 1)
    3 ^9 x* x9 ]6 l1 Z% Q: TF67 V9 o3 z  g7 z
    0 j6 h9 d6 N8 P( ~2 l$ g" ~/ m
    (x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
    . s" j- e  j; E" L3 C. K2 D. ^& M  w. m; K7 w( y( Q- G# Y4 c
    Q<x> := QuadraticField(6);Q;  }- d: Q  w4 J' r/ _- o
    C:=CyclotomicField(6);C;
    " Z! W* x$ O% m) B* r5 n0 HFF:=CyclotomicPolynomial(6);FF;
    4 W5 \0 \+ M% p* e4 ]: {3 t' K" l' a4 Q
    F := QuadraticField(6);
    ; ]9 s1 u7 S9 h$ c$ n- z3 ?% u% OF;) w; r; a& Z- B" ?
    D:=Factorization(FF) ;D;
    ) A4 V  {0 i$ _5 u4 [+ a0 @Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    . H/ j5 O$ Y& e6 a" A" d, ICyclotomic Field of order 6 and degree 2
    5 \' Y: j: m# t) K6 F* p6 L$.1^2 - $.1 + 1# r0 q9 ?9 {# q; q; B$ L  o, F6 w
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    5 Y# M; B: Z6 z* _8 f3 e/ g[& N% Z2 m) y; a- \
        <$.1^2 - $.1 + 1, 1>: X, ]* ?5 V/ B& [
    ]% c( ~/ q3 e, v% ^# n3 c& c
    $ P* c0 D  J/ L9 ?$ q* \
    R.<x> = QQ[]
    0 o# _- ~" ]0 J& PF5 = factor(x^10 - 1)
    , q( G9 |8 `7 |F50 `1 g5 B. F! u" C; Z  Z/ y
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
    . n# {! r6 B$ ~& q+ I4 N1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
    0 z3 \5 \* O& A+ Y. S% }7 r3 Y, j) e% h3 O' i
    Q<x> := QuadraticField(10);Q;# S" X8 E7 f+ x% r" J6 A2 C
    C:=CyclotomicField(10);C;
    1 c9 _" p3 v$ T: kFF:=CyclotomicPolynomial(10);FF;
    # Q, @9 q! Y! W' L4 z5 ^# N! N) V8 v
    F := QuadraticField(10);
    ! ?: g- L0 z: e2 x% T) ?F;
    : h  |  f3 l& Y) `: `D:=Factorization(FF) ;D;
    7 l( e+ k' N# D& @" wQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    3 l: Y* g1 I2 M# z' GCyclotomic Field of order 10 and degree 4
    $ L5 n( G2 i( p: U; w# M# q* {$.1^4 - $.1^3 + $.1^2 - $.1 + 1
    * o% T' a. P  `0 g( c8 \% cQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    - g$ |6 ^- [7 N# ]1 F[
    7 c+ A7 B4 N7 K& w0 N8 q    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
    # T- o/ C+ U/ b9 R9 C+ `7 O]

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    lilianjie1        

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  • TA的每日心情
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    2012-1-13 11:49

    • 签到天数: 9 天

    [LV.3]偶尔看看II
    10#
    发表于 2012-1-10 11:55 |只看该作者 |招呼Ta 关注Ta
    分圆域：123

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