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虚二次域例两(-5/50)

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lilianjie        

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    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 |正序浏览
    |招呼Ta 关注Ta
    本帖最后由 lilianjie 于 2012-1-4 17:54 编辑 : s* s& {& K+ _& c
    4 J% ]* R# I' a
    Q5:=QuadraticField(-5) ;
    8 c! S- ~- s3 L4 F) Q$ J0 sQ5;9 d2 u& C0 ]9 U" T- _
    / B. a( j1 l; C9 a& x
    Q<w> :=PolynomialRing(Q5);Q;
    ! [0 R: h. G3 ]EquationOrder(Q5);
    & C, }' W* w5 J/ r6 {* ~1 RM:=MaximalOrder(Q5) ;/ [/ i  C+ @- V' F; h7 r5 M" ?
    M;
    5 K( O) y! f4 ~& k0 y6 Q. QNumberField(M);5 A/ p/ \1 i- c+ N' M
    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;
    4 u% `7 |1 @$ r% T9 J$ \2 pIsQuadratic(Q5);4 u3 g; S  {2 C# K2 V+ u# `
    IsQuadratic(S1);
    + ^% ?& {( t4 v* pIsQuadratic(S4);5 m' q3 F  y. S( H4 v
    IsQuadratic(S25);
    ! U2 |3 G" z8 U; J6 BIsQuadratic(S625888888);
    " m# l. M8 g& f( _1 b) y7 {4 yFactorization(w^2+5);  
    / i. ^; [( Y0 Z! d  [! EDiscriminant(Q5) ;
    ) L4 @* c: I, `FundamentalUnit(Q5) ;0 W) y4 Y( s; [7 K
    FundamentalUnit(M);  _- L- Q0 s. h/ Y! V& c
    Conductor(Q5) ;
    4 ^; s% q" }) b  J; r# C% K, I& R' I5 N+ m' F
    Name(M, -5);
    # c8 Y1 f4 r; O+ g7 o$ SConductor(M);
    2 n2 t' v* w) g: S- eClassGroup(Q5) ;
    5 s& m$ [1 D% o2 s9 EClassGroup(M);( |) p' N0 {8 |3 X/ w9 q: t* Q
    ClassNumber(Q5) ;4 ]5 b0 M3 }: c( U1 X# B
    ClassNumber(M) ;
      K+ W9 E' }+ z* H+ U( W+ APicardGroup(M) ;
    4 F4 E. a+ M' c0 j+ KPicardNumber(M) ;
    6 W, U* o: b; |' {& g7 c& X/ v9 F$ B6 {9 W" g; U) z2 s& E. x
    QuadraticClassGroupTwoPart(Q5);
    " d2 E  v! ?1 D% N# iQuadraticClassGroupTwoPart(M);
    * y# b1 h& ]9 ANormEquation(Q5, -5) ;# n$ r, F5 Y" f9 v+ X
    NormEquation(M, -5) ;& n; g/ J& G+ u
    Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field/ Q' h  n0 k5 _$ i( u
    Univariate Polynomial Ring in w over Q5
    ( J2 v. m+ `# \. `Equation Order of conductor 1 in Q54 Z, [. @, ^7 a; o! a7 m
    Maximal Equation Order of Q5
    + V) l: s6 y8 r# ]6 u- u4 x3 lQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
    9 |: ?* q! Y( [9 p  iOrder of conductor 625888888 in Q5( K- W+ S, }3 N4 Q$ z/ i0 i9 B
    true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field1 Z/ F( O. d6 P5 N
    true Maximal Equation Order of Q52 x0 O" `. t6 x1 o. ~( g/ `4 Q
    true Order of conductor 1 in Q56 X2 G# i  Q/ L. o8 \6 A
    true Order of conductor 1 in Q5; J, p( k4 J) G! C
    true Order of conductor 1 in Q5
    0 c. ^1 R6 v, v: Z( k& R. @[9 Z. y! Y5 s+ t: m$ S/ Q. u) d
        <w - Q5.1, 1>,
    / U! u% P& X7 z$ m    <w + Q5.1, 1>
    ! x2 G7 p6 y/ ~$ E- b]" g5 ?# {* {& B2 U; t9 `7 h
    -20
    ' }+ W, r0 Q9 j, I! e6 M. Y
    * y5 e6 C. c2 h; I1 v9 R2 n5 Z& D>> FundamentalUnit(Q5) ;6 B* P2 M, L  C& d/ \5 Z
                      ^
    # y3 Z& T( R! x$ i1 @" ERuntime error in 'FundamentalUnit': Field must have positive discriminant
    0 M6 x* S0 w4 h& X( u
    + ?' L9 A$ O; b! ?, E" {& r5 n  T
    7 q" A+ [) F2 I6 s4 y9 _>> FundamentalUnit(M);
    ( U; Q' ]( D* Z4 l. _" }                  ^
    ) n! P0 y# d. ?* ?: QRuntime error in 'FundamentalUnit': Field must have positive discriminant
      f" c5 G+ V$ [5 Q( u% A9 |+ x2 m' c" i# @4 f
    20
    9 U3 q# U3 {6 F) M6 u/ |
    , H" q; K2 w& ^+ Z>> Name(M, -5);
    * K* Q* E; z) i$ c# @2 W       ^; H2 Z; Y! |1 S, L( ?/ V
    Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]' j! O2 o) P) K+ w3 v) ]4 h

    ) U% E  s: b: c4 Q6 D/ P- @- G1
    , f6 _) b3 a! @2 e$ L/ U7 h7 xAbelian Group isomorphic to Z/2
    3 ?! _% k: z7 G9 j$ E; MDefined on 1 generator" A$ O. E: K2 c
    Relations:8 e2 g' m, h9 D" H+ K; k. H* j, b
        2*$.1 = 0* K+ o  c- u3 _. c3 U
    Mapping from: Abelian Group isomorphic to Z/20 a9 s5 {3 |. o' H# ^. \- l; `% P
    Defined on 1 generator
    , {9 z5 T# Y6 F$ GRelations:
    ! z1 g$ W2 g, `6 U4 ^    2*$.1 = 0 to Set of ideals of M
      l) r) B3 a0 o! s( _Abelian Group isomorphic to Z/2& j6 D9 ^5 C# A8 b# t
    Defined on 1 generator+ |- O/ j( K; s- o6 ]% ~
    Relations:
    0 n8 {& L7 p# N* Q' e( n6 \    2*$.1 = 09 F% j6 W7 }: h. k; X* L& ^1 W
    Mapping from: Abelian Group isomorphic to Z/2
    ' `2 f. e7 t6 D+ ]6 sDefined on 1 generator, ]2 K' _) }4 p! v* l3 ~" c
    Relations:
    " {1 }. U2 D0 R! r% }* K0 z    2*$.1 = 0 to Set of ideals of M
    1 {$ z3 X2 `( a$ a- ^2
    * y2 o8 y' G/ I" V5 o/ @; u/ A" @1 y2+ k, V9 |. j( z- q
    Abelian Group isomorphic to Z/2
    ! A. ^0 J4 ?( h6 C' k: @  SDefined on 1 generator9 }4 }% T: X1 }1 a" r
    Relations:& N4 P# T+ d8 e/ @9 V. [
        2*$.1 = 0
    6 a  @4 ]' e$ _( I. o/ l( ZMapping from: Abelian Group isomorphic to Z/2
      B+ w1 I/ U+ @; I+ @; pDefined on 1 generator, u1 V4 B1 e% n- F
    Relations:
    " v. j  G7 `. X1 N! X. T    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
    ; y" \1 _3 G  v5 F2
    # C" I' v. L1 E+ PAbelian Group isomorphic to Z/2
    + d: y* R/ l( U: {8 ODefined on 1 generator( G- P* Y: {4 L$ l0 A2 B) g
    Relations:% ]4 P0 ~( N# F1 b6 s$ _; w$ j
        2*$.1 = 0
    1 r  T* F: N6 W% IMapping from: Abelian Group isomorphic to Z/2
    , w; ]7 I" m4 P  c. S6 eDefined on 1 generator' {7 Q& \% w$ t  p8 e$ Q
    Relations:2 W! y3 v% f% M2 P9 A/ \
        2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
    3 d2 h: o7 H/ xinverse]
    ' U/ Q; u; u7 I6 H2 P+ o0 o" nAbelian Group isomorphic to Z/2. _! H, ?7 H' @
    Defined on 1 generator5 \0 L. @, S1 i/ @: R
    Relations:2 C1 h" t2 [; |$ `& ]5 @
        2*$.1 = 0
      c& j. b7 J/ s3 jMapping from: Abelian Group isomorphic to Z/2$ }9 A) c2 P# o; q$ k; v
    Defined on 1 generator
    3 W" s% M8 t% K0 E* R" M# qRelations:
    . L& B0 O8 X! q$ K, e3 c    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
    4 ~! q/ V' T9 ginverse]
    # \& R- h+ Q' M/ `false
    0 l4 z( \9 v% s& s$ q' {! _2 lfalse
    7 q! u% C. H9 b# Y6 J, W1 w==============& I" C# Y4 \9 I$ A
    6 H& y' L( b# o
    5 z* T! R4 v4 z% D
    Q5:=QuadraticField(-50) ;2 |# T$ X. n7 w. S. F
    Q5;1 Z5 `. u8 ~+ k9 |2 `
      T+ r0 R/ W* E+ b' @6 n; Y
    Q<w> :=PolynomialRing(Q5);Q;1 k/ [5 E/ y/ A- h. _: {% a( w0 T
    EquationOrder(Q5);
    ) S- E: N/ ~$ w# AM:=MaximalOrder(Q5) ;
    % g- A$ t+ }9 t! m) |M;
    : j. M+ D; e8 v/ a$ k% |8 [NumberField(M);9 c& I. ?% w3 A( z& v4 M4 L  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;
    6 p0 W4 V6 y! E9 x2 p1 d5 e5 d, ~IsQuadratic(Q5);* `5 t) m5 K7 j5 }6 G
    IsQuadratic(S1);
    4 e8 N8 r" ^# h' M- q7 `4 tIsQuadratic(S4);
    2 x6 ]8 K# P/ X7 K" B2 S+ R" x: p# {IsQuadratic(S25);; Z. s2 B7 L# z! J% ?3 h
    IsQuadratic(S625888888);
    + y% U4 c5 U& D' r" r% _! a; xFactorization(w^2+50);  
    4 B2 R" G5 d8 _9 H, q% _Discriminant(Q5) ;
    0 g, v& ~* w) e: PFundamentalUnit(Q5) ;
    8 @4 j* u; l' j* [: h) U: ]8 v1 L, cFundamentalUnit(M);4 }  ~( R" k. s% }% j
    Conductor(Q5) ;+ u$ s" D+ _# F  y) T# g- O$ V% u

    " x5 b) m/ f8 g/ TName(M, -50);; x) k5 z" K3 m$ k0 E) J$ E
    Conductor(M);
    : n/ E, `7 ~3 s$ x/ I  _ClassGroup(Q5) ; : U9 g4 N) V+ s4 e
    ClassGroup(M);
    # y2 ^) I0 g1 S6 g5 [; [2 HClassNumber(Q5) ;
    . p6 \6 x0 a  }- P% VClassNumber(M) ;8 g3 b9 \/ S) N3 w6 F5 p+ m: X
    PicardGroup(M) ;; o# c# b& W: v4 ^+ G$ q5 H& `+ m. w
    PicardNumber(M) ;
    8 d$ `1 o# N( ?' n; k
    0 \( k) z) J* |' V$ t6 dQuadraticClassGroupTwoPart(Q5);
    " w& a; ?: i9 Z/ F9 rQuadraticClassGroupTwoPart(M);0 J  P5 C0 n, g9 P- t
    NormEquation(Q5, -50) ;
    ' e# |+ v8 f0 k% P6 Y1 A/ ?NormEquation(M, -50) ;7 ^5 s& R: k( C3 ?' |/ G) `0 m

    ' v3 ~4 C% j. @9 [/ g4 ]2 z1 l  h( sQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field5 W! ~3 _' H- Z* D2 @
    Univariate Polynomial Ring in w over Q5% |. _% C8 [" P( e0 M4 y5 t5 I
    Equation Order of conductor 1 in Q5& e; j; {2 L. v2 e
    Maximal Equation Order of Q5
    5 t/ T0 H2 L/ fQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
    . N6 }" v( Y9 b/ IOrder of conductor 625888888 in Q5) Z0 I0 e" r) Z( c* S" ^1 Y8 g
    true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field% X' `7 d( ?4 ], l9 Z: z
    true Maximal Equation Order of Q54 T* c; p8 y% D* ^, n
    true Order of conductor 1 in Q5
    # H+ r  v& p: L9 Rtrue Order of conductor 1 in Q5/ b' j4 f7 ~% J
    true Order of conductor 1 in Q5$ ?' t; y/ @* l1 s3 v1 i. q" u/ b$ ]
    [
    & _+ o2 B2 x& B. c/ m! y    <w - 5*Q5.1, 1>,
    , ]: L, M6 F2 x& F( }' X    <w + 5*Q5.1, 1>. x2 d0 x' E# I1 O$ d8 w
    ]2 L3 [/ ~6 Y  W& P7 a# b. `
    -8
    6 z* n, R* s, V1 E3 K: g! R1 C" K, ?" m# P. C  f4 Z
    >> FundamentalUnit(Q5) ;0 B4 c. C3 M! \
                      ^
    % L+ m6 S  _9 w6 m8 c! l1 @! ~Runtime error in 'FundamentalUnit': Field must have positive discriminant
    8 @0 |' [) o; h. F2 I) G/ N: T" s8 |% I
    8 \* D. `4 s: N2 q. g
    >> FundamentalUnit(M);
    6 `& ^/ t* P1 P% d) z0 o                  ^+ E. }+ B5 u3 z; K- P4 ~% }
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    ; Q3 I2 h% m- s1 Y  j; W
    ; u' Y2 q' @, G% I$ [8+ h$ E/ H: Z8 \0 t4 H! _# ?, G) ~. a

    9 s* x, G4 t7 p7 y9 r- y>> Name(M, -50);
      a) t6 V( O& V0 m/ t       ^; ]$ k( ?& {6 Q/ g- r
    Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]! @9 b7 V- A# t; z9 \

    2 {# x3 s+ \8 r/ e. X& j+ a1$ F' u5 g7 {2 j) y/ T  @
    Abelian Group of order 1
    ! X/ c5 W  U/ I1 nMapping from: Abelian Group of order 1 to Set of ideals of M
    , F! @, w) o" fAbelian Group of order 1
    ! M! r- ?) m  c3 `4 PMapping from: Abelian Group of order 1 to Set of ideals of M
    / U- M3 O! |2 ^" x) _9 V8 {2 r16 m1 n. L  I# I, K8 m8 R: v: `$ f
    1, j9 x8 u8 ~% M3 a! G4 Z) I3 [- S
    Abelian Group of order 1
    0 x; P8 C7 R" z5 ]Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    ' e; ?* y6 ?2 T4 t& V  B1 r9 N0 _inverse]4 z* x% o" |/ \2 g* }# S" U
    1& \, @0 }/ {7 [3 h# m* q$ H$ C# X! [  }
    Abelian Group of order 1
    ) `. L7 A* u6 w3 R% \Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    / |. S7 L0 t  s( I-8 given by a rule [no inverse]# `! G7 I5 o9 |
    Abelian Group of order 1
    % V7 a/ F/ b8 d8 Q* D- ?2 @Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant0 [1 R! }& o# S5 ^' j
    -8 given by a rule [no inverse]. i3 D/ _9 a2 [" p3 m/ U$ C
    false7 {3 X! v% Q4 Y8 ~% z
    false+ @9 _) R8 w6 s7 p8 i
    zan
    转播转播0 分享淘帖0 分享分享0 收藏收藏0 支持支持0 反对反对0 微信微信

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    2012-1-13 11:49
  • 签到天数: 9 天

    [LV.3]偶尔看看II

    回复

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    lilianjie        

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    2012-1-13 11:05
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    [LV.4]偶尔看看III

    本帖最后由 lilianjie 于 2012-1-10 17:41 编辑 3 V* @8 C/ I8 ^& ~! J$ b
    lilianjie 发表于 2012-1-9 20:44 , N6 p9 m+ b1 u) g2 r
    分圆域:
    # j' m" I) F7 [$ Y. hC:=CyclotomicField(5);C;
    ' Y) H* y% ~' Z3 f: uCyclotomicPolynomial(5);
    $ Q4 p$ V- I7 A6 _
    * h- F: w; N9 D; V) j2 `* Y- O
    分圆域:
    0 Q0 x% J. D) a. i分圆域:123* q0 y/ q7 A4 g

    " M7 p! L2 A  _) L5 U2 hR.<x> = Q[]/ [9 u% ~6 _3 @1 x
    F8 = factor(x^8 - 1)
    1 e' y& g& z1 U4 M6 aF8
    / N- ?9 P, M( Z/ M) E# |4 ?# @& b- _
    (x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1) . O8 x1 @( m. k" K8 k# A8 [1 h
    6 n' _: Z" C0 n: K* w
    Q<x> := QuadraticField(8);Q;
    ) C% U" E: R) E- Q: DC:=CyclotomicField(8);C;7 w' H( D9 }  \* n) L5 k3 [  t+ I
    FF:=CyclotomicPolynomial(8);FF;
    $ X1 t0 {- F# C3 w0 g3 b: b5 G5 T% k0 Q2 O
    F := QuadraticField(8);
    7 s* X" d) n' _/ @F;
    ! ]6 l: @" N: E! ~2 OD:=Factorization(FF) ;D;
    " B4 ?1 a  |) q% |6 B, b6 {Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    . E1 A0 Z& l- x* y; m' S1 MCyclotomic Field of order 8 and degree 4% {2 j- A8 z; Y) m
    $.1^4 + 1& H" H. z/ Y2 M, T+ l1 w/ a3 A
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field/ ~. b. T) J$ A! ]7 D* f
    [
    ( p6 N# c1 W$ V    <$.1^4 + 1, 1>
    / \6 Z, ?# N) s4 W7 }]5 Y7 ]! v1 I9 R- g
    $ N6 I1 T# {" ^  Z7 A
    R.<x> = QQ[]( r& E! x2 o- X  G: s0 t! r. y
    F6 = factor(x^6 - 1)
    ! F0 e; t8 q+ _/ q5 kF6$ e8 b; f1 b0 n; ^( s- ?5 |

    ) p( ]  W7 s) L/ j5 [3 ?/ K) Q. b(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)
    ; Q6 |# E$ V- p8 W2 Y! n8 K6 U. _, ~0 p+ M8 d/ E9 h  W5 b; e6 J$ @6 w
    Q<x> := QuadraticField(6);Q;2 v% H, Y8 `. N6 p: E7 s) T
    C:=CyclotomicField(6);C;
    " I  g) ?4 K$ J. j+ t' vFF:=CyclotomicPolynomial(6);FF;
    " N4 Z6 K$ F4 H9 H
    ! m' y/ U3 ]1 B) LF := QuadraticField(6);
    9 o; M$ z3 }' K) c5 |4 rF;. [; r: o4 m) J* A6 G& F( a$ c- ]
    D:=Factorization(FF) ;D;3 B* u9 S4 [6 |; G  B
    Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    1 _* b: ?2 A6 s5 e  r/ E3 uCyclotomic Field of order 6 and degree 2
    ) S4 J6 e( j! d+ g5 J$.1^2 - $.1 + 1
    ' @+ t4 [( ^4 KQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field$ ?0 }! `9 E5 Y9 J
    [6 w- N# I6 |! t8 r
        <$.1^2 - $.1 + 1, 1>* s/ n3 @% a5 i7 ^% c- c6 p
    ]* R8 E6 `6 @. l7 `5 Z# C
    4 G0 o0 n0 e/ V% r3 e- x6 e! h5 h
    R.<x> = QQ[]
    ( B, J* E; R+ L' Y! V. i0 h+ u, ?F5 = factor(x^10 - 1)
    6 }! Y7 y" i! w3 b9 J% a- ~F5: p% c# y1 ]9 ]
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
    + D4 Q6 i2 K; w0 T5 R1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
    % ?$ O. }6 O) P" J. J  _5 J/ P3 C
    Q<x> := QuadraticField(10);Q;
    ) @- y% V4 C/ iC:=CyclotomicField(10);C;
    ! ?5 a4 a" V2 K3 }6 C9 k2 o* UFF:=CyclotomicPolynomial(10);FF;
    * \8 e1 {. K! g  j* V0 ~6 W9 E1 P, ~; x) d- ?; D) Z
    F := QuadraticField(10);
    2 q; i) p' v: @( uF;
    1 ]% s  R) ^+ `# K& {% ?D:=Factorization(FF) ;D;% ?% K' |6 Z; a! D5 X6 I
    Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    , u' x( p: M; N% v1 tCyclotomic Field of order 10 and degree 4/ X, u& y; a9 }, @" L0 x" ]
    $.1^4 - $.1^3 + $.1^2 - $.1 + 1" Y: e8 J  x- y. L, h/ S! c
    Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    ) Y: i7 G9 s/ O* `( ^[
    7 V% U' q$ y; _6 z- c" t    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
    5 K9 J# l) _' F' X4 C* B/ j% K]

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    a.JPG

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    lilianjie        

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    2012-1-13 11:05
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    [LV.4]偶尔看看III

    本帖最后由 lilianjie 于 2012-1-10 11:23 编辑
    + u4 e- F) n$ \
    % ^6 y0 W! a& U4 O# `; A 11.JPG . C7 _+ R8 X* |. h: W" W

    4 ]& y  ^' ~9 V# L9 n- @( b. t+ m4 E 3212.JPG
    5 G* o+ O7 w( x. r7 Y( r! |4 X
    123.JPG
    9 ^- X3 b* l# T/ c  t8 x, D. j. }" h" ?# V) b# V6 R& {
    分圆域:
    ' T! K- L& L. V9 n' r" MC:=CyclotomicField(5);C;& K# ^7 C# U6 d4 V: k: U" x6 Y7 G
    CyclotomicPolynomial(5);
    7 l, A5 _/ F# O9 O) OC:=CyclotomicField(6);C;. H. J& A; s  W) s, J& Y7 |4 w4 }
    CyclotomicPolynomial(6);
    ( y2 `9 a, @; q/ ?CC:=CyclotomicField(7);CC;; q6 Q, L! X( J+ W
    CyclotomicPolynomial(7);
    - ~9 ?) q) t& J9 K2 X3 W6 b  FMinimalField(CC!7) ;+ |, y' |  p* o* M
    MinimalField(CC!8) ;
    $ W* Y+ o7 K) S) P# Y; ?$ }( [3 f+ OMinimalField(CC!9) ;
    4 j  W1 \' J' j/ r/ `# C( a* lMinimalCyclotomicField(CC!7) ;
    3 j+ K, J: \# L. {RootOfUnity(11);RootOfUnity(111);4 H% }) t# Y8 T9 A0 y8 b$ W
    Minimise(CC!123);
    $ g1 u$ X, s1 GConductor(CC) ;
    ; ?% b* S! Z! {/ t  ]+ h: JCyclotomicOrder(CC) ;
    9 I9 v/ B* }5 N0 a" H4 z7 e7 c+ d# C/ {* K3 V+ w) D' S
    CyclotomicAutomorphismGroup(CC) ;
    ; [- B- j. l% S* }
    " A7 [; H* ?! C" pCyclotomic Field of order 5 and degree 4! a4 J1 w* d- P1 t1 j" ^, \# h
    $.1^4 + $.1^3 + $.1^2 + $.1 + 1
    ' e/ V( _# B' I0 [$ yCyclotomic Field of order 6 and degree 2
    3 Z/ W& d  G0 l5 l# Z$.1^2 - $.1 + 1( W  p, n; }+ U" N
    Cyclotomic Field of order 7 and degree 6( z# n) _( H5 m/ Y; y
    $.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
    6 `2 d' ~9 T+ U* w1 n0 _* K0 tRational Field2 t  n6 y8 `6 v5 A6 F/ \
    Rational Field
    ) k# _2 ~4 y2 _, l1 _- a* a9 FRational Field0 e7 f8 X% O( L! |- v
    Rational Field- F& K9 \. I$ G' Q" A
    zeta_11
    ' i0 k9 y6 t9 Tzeta_111  L" x$ P6 @  @8 v4 Z+ ?' L
    123
    " `6 `8 m3 j6 }+ Z$ G1 z# d1 s71 y5 E3 S1 O( }2 }+ ^
    7
    7 F0 ~# k1 s0 g$ P- N, r' H. a( uPermutation group acting on a set of cardinality 6
    ! p1 z; F: P( AOrder = 6 = 2 * 3
    - q# U+ a: g) X# L- y    (1, 2)(3, 5)(4, 6)
      Z7 d( O: K: j. K: L" r    (1, 3, 6, 2, 5, 4)
    9 f7 y" A2 s  ~* f/ d) VMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of ! J) W- J; U; G5 c
    CC, |$ }) q) D" a( W5 M+ g. ^& W
    Composition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $, . _! E( x: Q: A# }4 r6 t
    Degree 6, Order 2 * 3 and
    * S6 P8 I$ Q9 W* pMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
    ) u. f9 @; o) D3 e& a; XCC
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    lilianjie        

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    2012-1-13 11:05
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    [LV.4]偶尔看看III

    本帖最后由 lilianjie 于 2012-1-9 20:30 编辑 ' ~% P3 p9 N+ ~. J+ @8 i7 n

    0 o; K0 x) ^0 gF := QuadraticField(NextPrime(5));) }) H5 H% W5 d9 \' T/ x. k
    : {5 x8 J1 t) _6 b3 L) ^. r
    KK := QuadraticField(7);KK;8 X* ]3 U9 z/ v3 K- w$ c1 S/ W5 K
    K:=MaximalOrder(KK);$ @  X) J. _. |  N
    Conductor(KK);
    4 q" [( H- R) T& H, ]+ o6 l4 R  e! dClassGroup(KK) ;
    , D6 C4 M2 s( [0 U# |QuadraticClassGroupTwoPart(KK) ;' r7 {/ a9 s) G. O9 J: L( C2 d
    NormEquation(F, 7);
    - |% k! Z8 m: Q/ U: V  @) o/ rA:=K!7;A;
    / b; i3 [2 s2 R" VB:=K!14;B;
    ' V0 x  G- D! t& i' [" |$ \$ wDiscriminant(KK)
    $ m' {( \2 F  t
    , r# j9 E9 t( u) jQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field* E8 L$ y0 N( i- `) q
    283 c4 W* w* @7 _
    Abelian Group of order 1. }/ K! o8 q3 n
    Mapping from: Abelian Group of order 1 to Set of ideals of K
    ' ^$ e6 R9 s9 W# y5 _Abelian Group isomorphic to Z/2- |) e4 j, W1 ?8 m" \' h  K/ i- f
    Defined on 1 generator
    ; S- O  m, l* `Relations:
    9 h* V8 p' R! u3 S" Z& v" V    2*$.1 = 0
    2 ~: }4 ~, K4 x0 D: IMapping from: Abelian Group isomorphic to Z/2
    9 l" w8 f9 j" r' ^0 v; Q8 aDefined on 1 generator
    ! w/ O+ N& p# B/ WRelations:
    + |# `) u' V$ S    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
    , K! H# T0 t! Q- a) |inverse]( P/ R4 s: T5 F3 h
    false1 [6 F; a% q/ H( \+ z& K8 T
    7
    9 X1 Z' }7 g# Y+ M1 s, h1 ?14+ e  W+ R. V' H. i- L( A
    28
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    2012-1-13 11:49
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    [LV.3]偶尔看看II

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    [LV.3]偶尔看看II

    Dirichlet character

    21.JPG (79.18 KB, 下载次数: 220)

    21.JPG

    11.JPG (74.76 KB, 下载次数: 228)

    11.JPG

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    [LV.4]偶尔看看III

    本帖最后由 lilianjie 于 2012-1-5 15:20 编辑
    ! l: a. }5 }$ E
      J: Q2 J2 G- ?8 a% |$ Q1 ]& v7 x' zDirichlet character
    : Q8 k4 j5 j, pDirichlet class number formula
    # _0 z7 D7 j( N: v  y0 i" X5 I* D1 U- M% B1 U
    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
    ( y$ t2 ]/ F5 W# F& {5 Z+ B! S* t8 z# L3 V% r4 r7 y$ u/ ^
    -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)]=1
    ' c' Q; y5 [) o3 k4 f
    : w8 n1 G+ @$ v. U2 h: Q9 _4 A-3时  6个单位根                             N=3  互素(1,2),    (Z/3Z)*------->C*         χ(1mod3)=1,χ(2mod3)=-1,2 |8 O, {, n5 H# F1 Q
    h=-6/(2*3)*Σ[1*1+(2*(-1)]=1
    3 p/ P* Z& \' g1 I# B+ q8 x+ [! P' N& t; d: A
    -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,$ Y, _2 \! u. ~$ [5 C1 q

    % @8 O! ?& i& d3 F6 O' d- u& E) i( ?' S  i# t
    1 Y0 W  S$ k; Y* x0 {7 N; ^
    h=2/(2*20)*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=20 R9 F3 k- A8 H, b+ {8 s3 o
    4 _' u9 n5 q  j, L) B* d9 p2 g
    # P: q+ ~( G# w& J! D7 |5 w3 U

    , ?/ g5 @$ _$ i3 f-50时  个单位根                          N=200
    8 P7 x' l: ]! p1 Z* }: w# a
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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

    群组小草的客厅

    群组数学建模

    群组LINGO

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    [LV.4]偶尔看看III

    看看-1.-3的两种:8 L/ ?) h- f+ C$ E" Q

    ! Y3 F* n0 x# ^5 [Q5:=QuadraticField(-1) ;
      ]3 N  D5 s% o) YQ5;
    1 W* V4 T+ r6 \
    4 F, x1 L) l# X0 N) K! nQ<w> :=PolynomialRing(Q5);Q;
    - O1 D4 {8 v1 K) `  @- FEquationOrder(Q5);1 F3 w( [; W2 r; V' Y
    M:=MaximalOrder(Q5) ;
    " o! Y, c! ~5 g9 E* dM;
    : Q* }% \7 T  wNumberField(M);
    & k; u3 X4 n7 FS1:=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;" E' a& T0 y$ q* H& D/ z6 C* G
    IsQuadratic(Q5);
    & d0 O. ^& K* p, F7 T' nIsQuadratic(S1);
    ) ~2 A2 e3 s& M( hIsQuadratic(S4);# W6 Y2 H5 m" D* }! G
    IsQuadratic(S25);
    ; P! g7 H9 I" i- IIsQuadratic(S625888888);
    9 x( V# I3 Q' q9 qFactorization(w^2+1);  
    # |* W# c; g" O( O: w1 k, SDiscriminant(Q5) ;% @$ ?* n4 P. N3 z
    FundamentalUnit(Q5) ;
    ) T2 ]' _7 q+ W. }7 Y/ b! l4 SFundamentalUnit(M);
    6 L' n" W  Q6 l( J" }4 GConductor(Q5) ;4 C6 G( c6 r. ^3 G) Z+ z/ N' \
    6 K* W1 s' g1 I/ p
    Name(M, -1);/ P& l, {2 H- A" h
    Conductor(M);, [; t$ M% `, ]$ _  P6 w8 r
    ClassGroup(Q5) ; # H9 Y0 G2 }- T9 L* @) c
    ClassGroup(M);" @1 u& W+ ~7 k
    ClassNumber(Q5) ;
      D- G  ?9 G+ A* |2 k! ]ClassNumber(M) ;
    ( ]9 _" O" r2 s$ b: \PicardGroup(M) ;9 ^: i: J/ S2 p& J4 [, `. o' z. B
    PicardNumber(M) ;/ U6 C$ F5 ~, O4 U# i+ ^6 r9 p

      {; m  [1 V9 D: C% f: ^QuadraticClassGroupTwoPart(Q5);
    * I3 ^# ~9 N. }- `! a9 K4 g+ }0 KQuadraticClassGroupTwoPart(M);
    4 w! {4 d. g* O4 @NormEquation(Q5, -1) ;% [! ]+ p$ \- t0 O$ B, p6 @3 D: o6 \
    NormEquation(M, -1) ;' Y- h( {+ P/ C5 U

    % p- E+ z7 X: w( X! Y' WQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field4 ]* N5 O, B4 F, R4 U3 U
    Univariate Polynomial Ring in w over Q5) M6 Q# C% O0 K2 E
    Equation Order of conductor 1 in Q5( K) F: o  u# ]- h0 A# E) c- r. q# X+ i
    Maximal Equation Order of Q5
    + U" }7 {" N3 _: W& P+ ]  wQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
    ' j# e5 y: j3 jOrder of conductor 625888888 in Q5# ?2 N& z0 Q# H$ ?' O. Q
    true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field9 G# C! v6 z+ b
    true Maximal Equation Order of Q5
    ) r" v8 E0 k8 c" S  Z6 Itrue Order of conductor 1 in Q5
    % g# X/ E4 G* utrue Order of conductor 1 in Q54 F1 x* E7 \* q. t) U9 t! p4 f- a
    true Order of conductor 1 in Q53 E0 ]9 u6 s; o) n  I* ~8 W5 ^
    [. {: V2 U6 T7 A
        <w - Q5.1, 1>,  Z! R; a3 {- q6 L
        <w + Q5.1, 1>
    0 c  g- Q2 x8 m$ J$ |) Q]
    ; Y, a* i' Q8 q$ ~-4; l7 D! w  n# y8 O  J! S

    ( O. R& X; ~  B9 C  _>> FundamentalUnit(Q5) ;+ H! N* U" @" X
                      ^+ R& a' Q) P% |( `5 d
    Runtime error in 'FundamentalUnit': Field must have positive discriminant0 ]9 v# y! U8 h# a% G2 \) H4 b7 u
    + T  I- X3 X* y7 D: w
    / \9 Q( C" k5 }  e
    >> FundamentalUnit(M);
    * Q4 J/ i% m2 N" f! p, K                  ^& g8 C/ E' Z, ]
    Runtime error in 'FundamentalUnit': Field must have positive discriminant
    1 G% o/ K* `6 X& K; Q9 w
    ! j  r9 X" L5 ?2 B43 V; X/ `$ @# J; ^' S7 q
    : Y. j! o4 ]/ `
    >> Name(M, -1);
    9 c0 i& w/ g5 ^% U' g( [2 T# u       ^6 |% g4 r5 J5 L) m8 k
    Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]* t$ \( ]* m$ y

    " q6 P& O; l7 g1
    2 `# j6 ?  d  b" mAbelian Group of order 1
    9 q3 |. S* u5 i$ C$ q  P; qMapping from: Abelian Group of order 1 to Set of ideals of M9 z" L1 i, E% X0 U( o0 C( H
    Abelian Group of order 1  i. ^1 Q; O5 j4 s
    Mapping from: Abelian Group of order 1 to Set of ideals of M7 ]9 }+ g  W" z4 l8 }% B
    1" v3 O/ j3 \4 w+ w( b- a" K7 p
    15 s5 p! H! A3 k/ K- ]) I6 L
    Abelian Group of order 19 o9 `$ d4 Y4 l$ @
    Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no+ d# _- \3 p4 U; w
    inverse]
    5 L6 J/ D/ v7 I( ^* e0 |14 g3 ?; p8 f: \! t  L: @. b
    Abelian Group of order 1. O/ @4 ~! {  [# V
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    ( w2 o1 A* |. m3 e, I, f-4 given by a rule [no inverse]7 R3 h% ], K2 `2 y1 u+ ^
    Abelian Group of order 1
    0 R, E8 d6 I, {+ O, ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    - H0 z* X9 I& t0 K0 U1 J9 P: P-4 given by a rule [no inverse]+ O$ @2 B$ n6 d
    false* k+ s$ |/ Z& R6 w2 b
    false  a- H( w' x0 M8 |+ C( c( q8 R* m
    ===============( M" F8 {, r4 c

    6 g* R1 b; `% @0 d% l7 ?- C" i: IQ5:=QuadraticField(-3) ;& B0 [% T5 K  F* K( p
    Q5;: _7 F# }7 d" s+ y& f
    6 ^8 D* J$ v1 d6 G$ W
    Q<w> :=PolynomialRing(Q5);Q;/ M  k9 B) Y+ c, X& Z; g! [7 I
    EquationOrder(Q5);
    , M$ q" L% q$ u) c- OM:=MaximalOrder(Q5) ;0 I( j7 A' z+ H1 h3 [
    M;! N: J5 q0 e% w1 j7 R
    NumberField(M);. V* E9 R3 v) b* n
    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;
    , S6 ~5 z6 `1 q8 V0 m2 RIsQuadratic(Q5);' Y0 n9 \! M3 t& w7 F6 W+ G* V
    IsQuadratic(S1);
    0 t9 v" d8 O4 `4 }9 M4 Z9 {3 hIsQuadratic(S4);
    5 ^2 m8 k4 D* Z3 a+ w4 \IsQuadratic(S25);
    , x9 o3 J; ~9 G) d- N% PIsQuadratic(S625888888);5 l  }( M. n4 e  l6 |" J
    Factorization(w^2+3);  
      m" G% {8 S, E3 f6 A  DDiscriminant(Q5) ;4 z: ]4 m, Q9 j
    FundamentalUnit(Q5) ;& {. E+ j! ~8 L# O- N1 R& p
    FundamentalUnit(M);- E7 L$ [# y( m( `5 r+ I
    Conductor(Q5) ;4 Q: j# O' }) u( @6 n+ u

    # S6 d: @. m5 k$ S: OName(M, -3);
    . y  {5 W; f* K3 uConductor(M);
    # I& X* Z, e1 _0 ~7 W' g3 v* OClassGroup(Q5) ;
    4 y3 q6 Z) f. i' P5 H$ lClassGroup(M);
    - e1 k# P( M  S; d9 `2 X3 yClassNumber(Q5) ;
    # w3 g8 n; n5 X1 s+ ~/ `6 _ClassNumber(M) ;+ _' R; s4 G. U% i5 n  ^( N: M
    PicardGroup(M) ;3 R# u4 W  N+ q# m) n6 f# d1 ]/ [) E
    PicardNumber(M) ;
    " c7 K3 R" ?5 V3 e" ]5 `( A. L- E
    QuadraticClassGroupTwoPart(Q5);' j3 i: E$ ]6 ?" ]; `( j* G
    QuadraticClassGroupTwoPart(M);% P% M, C/ @9 z. t" g3 v( p
    NormEquation(Q5, -3) ;4 ^. r2 M) D/ I) v& _; s
    NormEquation(M, -3) ;& @- u# A$ S" h  C6 @7 ~9 S
    ! s; R. ]6 P. `- v8 F
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field' B7 \! S7 @8 M" n! z# [% s! d
    Univariate Polynomial Ring in w over Q58 N* f1 O/ J& z/ A, h, T
    Equation Order of conductor 2 in Q5
    # a4 n# l* B) B( T' q# q# NMaximal Order of Q5& F3 p( L# d+ D' `, z
    Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field; p* @* V0 G$ e+ v! C- ~# H
    Order of conductor 625888888 in Q5
    0 a6 Z( @2 D; F% o0 h# w1 ^true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
    4 x: F8 p' h! b4 U0 {2 Y7 Ltrue Maximal Order of Q5
    ) Z) o+ f9 j7 F! Y" j& |7 w8 x3 \6 Utrue Order of conductor 16 in Q5) E8 F$ W; e5 y) q- |
    true Order of conductor 625 in Q5
    ! V: H8 D! l( Dtrue Order of conductor 391736900121876544 in Q5
    7 P- A, J4 X- A( D" C& j, t+ Y* n[
    + d$ ~$ T5 t9 _  D! N2 q4 l* A    <w - Q5.1, 1>,
    $ C3 R8 [2 r) _* q! ^  P& y    <w + Q5.1, 1>+ V: \6 e4 C# q0 ?+ y
    ]
    & h: x5 S- {. O- [! T/ T-3
    . g9 ?# z- u& o% x" C' \- U: E; `( O2 \! a
    >> FundamentalUnit(Q5) ;& Y/ g& T0 G4 C/ h6 Z8 Q; n7 i
                      ^
    ' G$ \/ `  S) r5 Q7 N% }. gRuntime error in 'FundamentalUnit': Field must have positive discriminant5 a/ |( ?$ @: I4 K
    8 ?0 |0 G* {* {# s
    " l# I- _! G$ {2 Y( v2 X
    >> FundamentalUnit(M);
    9 t( q' B  C6 p7 Y                  ^) J9 R/ K4 ?% I4 o( ^! v' }
    Runtime error in 'FundamentalUnit': Field must have positive discriminant6 I/ g+ T% A2 i! w0 x

    , V) L. |# d6 `% {  S36 ]- `; ]# X1 }3 ^* L& t' h8 g
    & A9 A- e$ T- ^" f& [3 U* f- ~
    >> Name(M, -3);" t  y1 y" o5 J; q9 D- I  q
           ^
    9 n2 Y. f/ @- ?$ x: Y6 T( bRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]0 Y2 [8 n' I/ P# U0 J

    9 J  ?: l/ Y0 ]$ H) K1
    6 ?% k( I: i6 p; l# UAbelian Group of order 1; h$ t' Y0 k! G$ M
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    2 V& T: w8 F% ~4 n7 r1 lAbelian Group of order 1
    . [$ c0 r- w* O* s9 k0 n$ NMapping from: Abelian Group of order 1 to Set of ideals of M9 D$ b. z" }1 f$ U% R3 V# f: t
    1
    4 |, C/ \! q. j% V. b& x1
    & w% o- X, E6 K( m1 p& n/ jAbelian Group of order 1
    ) p/ J- X6 Q) ?6 _4 kMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no2 B) }/ ^* Z% T4 k) Y3 z
    inverse]! @2 ~% K% R0 ~
    1
    7 x5 N5 V5 n- h5 z: }+ I# DAbelian Group of order 19 w4 d2 R5 X2 S+ W) {9 K
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    $ e3 q5 a, g. ^0 x-3 given by a rule [no inverse]* T2 X% ?5 H0 s$ [& z+ {% \
    Abelian Group of order 1
    ' a$ Q( D- I7 n& i3 y3 e. lMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
    0 g' r4 u) X; Y" |) G- D- Z-3 given by a rule [no inverse]
    0 u/ ?3 M$ i+ [/ w- q% jfalse
    ! A$ c) ?* O  p5 U' Pfalse
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