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实二次域(5/50)例2

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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

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    发表于 2012-1-4 14:05 |只看该作者 |倒序浏览
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    本帖最后由 lilianjie 于 2012-1-4 17:59 编辑 1 W; H% ]+ W; J# ^! ^( O1 o

    9 X0 C% b9 A: N5 A7 t: I0 u9 oQ5:=QuadraticField(5) ;( Y: b* d2 s  @! L1 x
    Q5;$ @8 c: M, `% [, Q
    Q<w> :=PolynomialRing(Q5);Q;  Z4 K, ]: w" {

    # `8 D* E- O7 S2 dEquationOrder(Q5);
    3 w) U( ~+ L, ?M:=MaximalOrder(Q5) ;
    2 y, ]5 O- L5 [  t9 i( \5 H# hM;
    3 s, U6 U* o/ J1 x1 r6 b: hNumberField(M);- D& c8 h6 d6 X' Q$ F; ]& X
    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;
    / a& e, p0 ~9 qIsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888);5 G/ q2 d  b3 E9 v7 `0 `1 H
    Factorization(w^2-3);
    & v5 Y: ~6 }; a; t4 ^9 QDiscriminant(Q5) ;
    # |! k* }1 I9 t# Z5 ^' @' d. a( [FundamentalUnit(Q5) ;* W# G0 l$ @3 m& {1 `
    FundamentalUnit(M);; a6 }) C) v2 u3 r/ O6 B
    Conductor(Q5) ;
    9 \) W: I8 H' a6 k  |" T$ H+ a+ x. |Name(Q5, 1);7 g: k1 s: M. I
    Name(M, 1);
    " U  Z1 e! g' b% \$ {) [, o7 z! JConductor(M);3 ?  N; p! Y8 Y& }5 b3 G
    ClassGroup(Q5) ;
    2 U3 y+ y! ~$ |7 `/ l- n: A" S1 nClassGroup(M);* j, @2 @+ d4 J* X* W
    ClassNumber(Q5) ;
    ) v4 D1 G7 R! |$ i: f2 AClassNumber(M) ;! }* t/ Z8 T3 d

    2 S- g1 \: A4 q( ?, \! gPicardGroup(M) ;7 @& i2 _0 R! ^  o  M
    PicardNumber(M) ;( m' B2 l  \5 O9 x
    ) {" B: u$ _% O, z2 ]0 _

      N4 V: _9 _" b9 ]. M" OQuadraticClassGroupTwoPart(Q5);$ @, G8 `  C0 H% @9 P% Z1 m, U
    QuadraticClassGroupTwoPart(M);
    9 @! z! Q, O* ^5 I. U6 ?. W" I  E! {" E$ U+ m7 }( Y% b: e7 `
    ) \3 `/ c) k& g+ C( N* \
    NormEquation(Q5, 5) ;
    ; @- n7 V3 K- ]NormEquation(M, 5) ;
    ! j6 O; d' e: k, p4 t8 G$ s/ M4 [1 g% H" s) _* H
    ; W/ J) O! U& {$ \' b7 x$ G
    Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field
    % k  d) r; N/ W7 r1 oUnivariate Polynomial Ring in w over Q5
    1 ?4 |3 k3 ?. J0 J6 C) j; REquation Order of conductor 2 in Q5) k  T. c' z* V; `# S% P4 d
    Maximal Order of Q5' G# t$ Q6 e  l" {, q* L/ E/ N
    Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field/ x6 v$ B0 d6 W  z9 H5 m% Z! k- l
    Order of conductor 625888888 in Q5
    " Y. M9 ^! U' _* A8 a- etrue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field6 |0 L" U2 t6 B8 Y4 g' `
    true Maximal Order of Q5
    ; k* f6 B/ G0 z& ^: Q  H1 U# w& Utrue Order of conductor 16 in Q5% x% I. O! l) @) d5 n+ ~# t! G
    true Order of conductor 625 in Q5! f+ o  `' ]' w2 x/ @( ]
    true Order of conductor 391736900121876544 in Q5
    $ ]: f8 v/ h3 v( w[! F* _) \$ C8 `: C8 y7 i" E+ J7 ]
        <w^2 - 3, 1>
    ( Q0 Z8 q. m9 G  _]$ e* C) Y+ ]! p, l
    5
    + I8 n* E1 m& g) f1/2*(-Q5.1 + 1)) |# G. }5 Q. D  u. v
    -$.2 + 1/ Z  r* @, U! d
    5& o% U* k) p, F' G* ]! w6 ]
    Q5.13 g# I( B6 p8 |$ N: O
    $.2
    / L% Q( s0 s3 ~+ r: A4 o: u1* b) Q* t2 w0 _7 o- n* o
    Abelian Group of order 17 `. p& w4 B  \/ l
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    0 n& \1 C* `- l1 L, c+ D% oAbelian Group of order 1
    1 M* p1 p! M* `( N6 c& m& g# `+ u( W% ?Mapping from: Abelian Group of order 1 to Set of ideals of M/ P! C: ^( D, D; V- ?8 T" T
    1
    + s+ W2 q+ j. j/ }, \1
    1 Z! G& }7 d) k6 H" sAbelian Group of order 1
    - S, ]- h( Q) a6 O3 AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    - T, N% ^0 l3 |* m. P1 k; D/ c2 ginverse]& @1 [% ?7 T$ i" N
    1% Q7 {) W, Z5 v
    Abelian Group of order 1
    " x: V) ?9 }7 H6 Y1 `! Q# jMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant8 b( Y. y- l2 x( ]
    5 given by a rule [no inverse]! w. a0 z* g7 p  {
    Abelian Group of order 1# M, y) h* H0 s4 r9 X
    Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant$ x& c; ^# D6 e9 n- A# ?6 A' E
    5 given by a rule [no inverse]1 m% ]6 g$ r+ H( G0 I- y) [1 U
    true [ 1/2*(Q5.1 + 5) ]
    & e1 q1 }1 w8 O$ _1 i% ctrue [ -2*$.2 + 1 ]* M3 P6 l8 R7 Y. I' {' F1 C

    1 y- b) n; [% _# O/ A5 a! }/ R1 B
    1 H8 k: j, C3 ?/ _. e) k; Q* R
    5 j  @3 ]- C# S) P9 ?0 x& t! C# }3 z+ z7 O) @9 Y. L% @* x# W! P

    # [0 j. j: J) Y% `; l
    1 Y8 g2 v( T; [3 V
    1 |5 P  v6 d& [# m+ B# R7 M9 Y0 X9 z9 n5 p( v! v- o
    ! [. |7 C* q! m& m3 q  Y

    & u/ r+ U, s% I$ G4 j
    0 B! W- @% ?$ e==============
    + _; |8 ?/ O+ Z$ P0 c( ^8 L1 ?' m/ t( x
    Q5:=QuadraticField(50) ;
    " C% r: s5 M  l* ?) |- R/ |Q5;  U  O4 G5 V- o2 c1 w5 J% p6 b

    * m& C0 Y! M  d# OQ<w> :=PolynomialRing(Q5);Q;
    . M( Y+ W( i& v- SEquationOrder(Q5);6 h% s- L0 [4 ~9 L; [
    M:=MaximalOrder(Q5) ;
    6 E: y7 t- M* ]M;9 J$ C& l! X: y; Y( P
    NumberField(M);7 K; ?9 y( y8 R) c  j1 }# \
    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;& x/ V' m2 r1 m- P
    IsQuadratic(Q5);, z8 ^: k+ f* j! Q* F4 Q  p3 }
    IsQuadratic(S1);
    0 F) E4 a- V- KIsQuadratic(S4);6 s( B1 z1 ~0 N, |! N* V: Y. F
    IsQuadratic(S25);: @+ B8 E' X6 ]2 p
    IsQuadratic(S625888888);
    & }5 B3 h# Z7 k# LFactorization(w^2-50);  
    3 A' O( N6 y. r0 C3 \, zDiscriminant(Q5) ;5 Z( {: v! s' U/ d7 d% A8 p3 |
    FundamentalUnit(Q5) ;
    ! C( @( d! y( yFundamentalUnit(M);
    . V7 E) N1 s" O5 \: y) {Conductor(Q5) ;: {: i" s# t$ Z; E4 Y' c5 P

    ; p* @6 p% }5 ]+ s+ V' bName(M, 50);% ?9 |3 ^: |  Q! Q- j7 ?! h
    Conductor(M);
    # [2 S% _! |, H  g9 KClassGroup(Q5) ; 1 N6 D1 J- ~. Q- z+ l4 O
    ClassGroup(M);2 ?& j) ]! b. a6 f
    ClassNumber(Q5) ;
    . U! t9 ~  ]7 T3 gClassNumber(M) ;
    2 c8 h, h  B$ D2 y7 kPicardGroup(M) ;
    $ O, W0 I7 r1 b2 @7 H/ P6 sPicardNumber(M) ;
    ' J& M$ O4 Y" `2 ~# R
    8 Q, x5 H7 m6 Z+ R: uQuadraticClassGroupTwoPart(Q5);
    % q4 r$ a  y* @& m* wQuadraticClassGroupTwoPart(M);+ l0 Y0 w; C) B1 H4 o6 \
    NormEquation(Q5, 50) ;: [' _0 G1 r  a& }' U
    NormEquation(M, 50) ;
    ( Q8 e" Z( E! @$ S% L% M1 F7 C- s& @: f% l: O
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    9 T$ R3 T( g8 b  zUnivariate Polynomial Ring in w over Q5
    " i+ ~/ @  M2 i' KEquation Order of conductor 1 in Q5
    " J# t) ]- K) p8 EMaximal Equation Order of Q5/ S4 }# ?' L( }  U8 o2 k; m3 o. I+ d
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field( ?* I3 b& U7 s
    Order of conductor 625888888 in Q5- q- y6 J$ d+ {6 A% B3 P
    true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field$ ]; @) J9 c7 O
    true Maximal Equation Order of Q5
    9 h" w3 X3 \& strue Order of conductor 1 in Q50 K0 t9 w# a, p9 b5 v5 I0 C7 n& Q
    true Order of conductor 1 in Q5& D0 K/ u  S" k7 L# J! i/ H
    true Order of conductor 1 in Q5
    7 C+ [! J0 M* q1 V& C4 a8 D* F/ q[" ^; D" y( l% G+ ?; e5 \9 l5 [0 w2 d2 W
        <w - 5*Q5.1, 1>,. [8 f3 e9 j: _+ h) A- Q. `0 ^1 ?
        <w + 5*Q5.1, 1>9 I" k' \, h4 @$ Y! @! q4 v+ b
    ]2 v$ _6 }9 a0 v2 v6 Q
    88 u6 @: ]3 \$ p4 o  e
    Q5.1 + 1
      U, o8 K0 q# e6 P& R& E0 }$.2 + 1
    ( A4 N4 z! U- v$ N* Y+ ^8
    ; w: l8 G. a4 O, a' A8 M3 e( H5 }7 I# v" y; t
    >> Name(M, 50);
    * Z8 F/ J! @& e1 ^, g       ^
    ! M. F7 m1 o" k/ B; DRuntime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]
    2 R% X& u! d- o8 J
    & G9 j" W% ?2 ^1 {; m+ ]& [# ~8 t" P( I1
    5 h' k: G( m8 j3 `Abelian Group of order 11 C- k8 p7 L8 m/ K
    Mapping from: Abelian Group of order 1 to Set of ideals of M
    1 ^- N5 P/ ?/ D- [1 h9 AAbelian Group of order 1
    ; e- e4 w# x- v+ _& L/ Z; {Mapping from: Abelian Group of order 1 to Set of ideals of M* T6 h0 q3 J8 {. t3 P
    10 G" t1 X6 y( w2 l( O4 p: }
    11 |. X  X6 ]. H7 t
    Abelian Group of order 1
      \& x- ^. p/ ~7 t# M6 `1 _Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
    6 R& Q6 t0 D8 z3 J: Cinverse]4 l* m% r' `: O5 ]& t) a
    1
    , O+ L6 Q5 ]0 P4 ~" E( dAbelian Group of order 1
    2 _; [1 s0 M( _) a$ dMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant/ f: W+ r6 I4 Q5 U. U
    8 given by a rule [no inverse]; Z3 H% I# M6 s
    Abelian Group of order 1
    & H% y& ^8 m# t5 F2 ?7 [* R2 JMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant2 c9 n2 k! @2 J# L1 d+ q
    8 given by a rule [no inverse]
    9 x! h- U1 F+ I  Ltrue [ 5*Q5.1 + 10 ]$ ?. @" N, @1 r! Y7 Y
    true [ -5*$.2 ]
    zan
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    lilianjie        

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    [LV.4]偶尔看看III

    二次域上的分歧理论

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    lilianjie        

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    [LV.4]偶尔看看III

    本帖最后由 lilianjie 于 2012-1-5 12:42 编辑 # b$ ^* j8 v1 I: a6 d

    8 O4 U" u$ M* |  N- S6 [+ v3 L基本单位计算fundamentalunit :
    7 N- A9 P! I) u6 }, ~8 H: O0 c5 mod4 =1                                              50 mod 4=2
    0 D1 k( v5 b2 d
    * o9 b9 Q- T) V x^2 - 5y^2 = -1.                                 x^2 - 50y^2 = 1.
    % q( G1 \& X3 F3 Q* D x^2 - 5y^2 = 1.                                  x^2 - 50y^2 = -1.
    + f' N7 C! X! a9 P' e& k ! @" W" M4 f  |( M! _6 [

    3 g0 d/ o3 c0 @+ n( C最小整解(±2,±1)                              最小整解(±7,±1)- L, R% ?! P) x
                                                                 ±7 MOD2=1* ?2 o5 e+ u# q2 X" W& G. A
    $ j( s9 L0 F& F) g9 w% U
    两个基本单位:

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    11.JPG

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    [LV.3]偶尔看看II

    lilianjie 发表于 2012-1-4 18:31
    : b2 k4 S, B( i, s0 F8 t+ D/ R基本单位fundamentalunit :$ A6 P7 l' l2 j* p% y5 I% A1 P+ E
    5 mod4 =1                              50 mod 4=2

    # j) v$ y; n5 z* d0 d基本单位fundamentalunit

    3.JPG (105.07 KB, 下载次数: 299)

    3.JPG

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    2.JPG

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    [LV.3]偶尔看看II

    本帖最后由 lilianjie1 于 2012-1-4 19:16 编辑
    / D1 i5 ~8 |8 l. j( g
    / A# \6 i- M) k( r# Y/ [判别式计算Discriminant8 ]( s/ {9 z1 n6 ]; a) Z- m! W

    2 d) d9 U2 z. Z8 [! E4 k5MOD 4=1 : x6 `+ V4 z" j" X  M

    1 N' X. {/ w4 }6 @: a6 ^8 c  n(1+1)/2=1          (1-1)/2=04 [' V4 q, ~7 A5 e
    / q% S  p8 E, q/ z3 K: Q- t- t
    D=53 H. j6 ]% Y$ t) Y! L+ h
    ; s9 k& G: J, F8 R* D4 y8 b( e/ f

    $ K: E8 i& F5 q  g2 J  a) |: u2 _50MOD 4=2+ n5 l5 |& ^, i5 c
    D=2*4=8

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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

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    lilianjie        

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    [LV.4]偶尔看看III

    lilianjie 发表于 2012-1-9 20:44
    7 G$ D- j( Z% z0 t5 z0 x+ E
    0 \& K. |! m, j/ }: b6 Q9 B* ?1 v  O分圆多项式总是原多项式因子:: ]. {6 R' N9 i$ X- N3 g9 n( P/ i
    C:=CyclotomicField(5);C;
    5 h6 g2 `8 l, ]; P, N5 @2 `CyclotomicPolynomial(5);
    ) V/ a7 r/ y' w! t

    3 D5 r9 e  x% S" \分圆域:/ U: G  k/ {% {
    分圆域:123
    & Z1 ]" b. {5 v8 b
    0 N" J' G2 J' \% w$ i7 P2 a8 h- bR.<x> = Q[]0 o) ~, u  T; i' G
    F8 = factor(x^8 - 1)
    ( I3 u5 K: t) k$ D9 ~4 s/ t% b/ HF8- F# l  f9 S, b
    9 T8 P2 y" z- w9 Q3 q% Y
    (x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
    * A: o7 c  x6 c+ v' l( E& R- v  @5 Z. D  t
    Q<x> := QuadraticField(8);Q;
    & M7 v# E% Y6 E$ |$ K1 h. }C:=CyclotomicField(8);C;
    * ]& H2 u) _3 S9 t* t, m$ kFF:=CyclotomicPolynomial(8);FF;: R+ D; x' Y5 e- s5 ?8 H
    & r& Z0 O( o0 J# l
    F := QuadraticField(8);6 n6 Q4 h3 n1 }9 A
    F;
    7 l+ k9 ]$ J( O- M: x" [: [+ ^D:=Factorization(FF) ;D;
    0 u8 M7 `* }4 C& }9 XQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field. R& v- I' y. P$ _
    Cyclotomic Field of order 8 and degree 4
    9 m" D4 V1 Q* Z1 l2 Z2 B- \$.1^4 + 18 ?( k" |! y( g$ H2 ?6 V
    Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
    % E/ L5 t  }, Z7 X5 S[
    , B* S4 c# I/ P( v% H+ ^    <$.1^4 + 1, 1>
    ( L9 S; n+ h; [1 x( R) O$ R]7 U* U, N( ?& t1 o+ l

    0 `3 w0 J5 _: NR.<x> = QQ[]
    2 ?/ t' e& k. PF6 = factor(x^6 - 1)7 B# [2 E7 _7 b, \# y* ~4 p
    F6; O6 a1 a* W( B6 m& _

    7 G' s# `4 R) \: [1 Q- Z0 ?(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1) 8 Q" ?3 W# ^. ~* Z
    3 o4 R) ^/ ^4 t7 X$ q5 }
    Q<x> := QuadraticField(6);Q;
    7 _7 j% P' y, B/ Q; EC:=CyclotomicField(6);C;
    8 Q6 r4 V1 T1 |) d6 {FF:=CyclotomicPolynomial(6);FF;3 ?; Q0 `1 l! m/ y
    5 U# w- l# s2 i: j' R# W2 c7 L0 X. `$ Y
    F := QuadraticField(6);8 g0 V3 I0 c0 w3 {
    F;
    % Q$ O% q0 b8 y+ s0 WD:=Factorization(FF) ;D;
    4 a4 g' j7 A" S) O. w# A9 x& y- NQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    0 {0 R6 [# m* X$ i5 I% N4 A' bCyclotomic Field of order 6 and degree 2  U" T0 E4 {6 s6 G8 z+ l
    $.1^2 - $.1 + 1
    * U' D% z1 D/ W/ RQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
    2 F4 c  s4 N7 @+ b[
    ' C$ D$ [4 U, N    <$.1^2 - $.1 + 1, 1>
    0 g0 g4 O4 m. @: y]
    4 ^( Y2 N/ L1 o$ i+ P0 }' D
    7 z- c0 r  a0 |/ l) L9 Z, j7 zR.<x> = QQ[]
    9 c! E6 L+ }$ SF5 = factor(x^10 - 1)
    2 y  u( r4 g2 q4 {5 k, Z1 yF59 y1 q" g0 Q( D: Z9 {! o
    (x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +9 t1 f3 m9 ^% K" g9 b* h
    1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
    " t! q6 D1 t: K* o2 Q# {3 _5 V1 X3 J: Y" S" H& x
    Q<x> := QuadraticField(10);Q;7 e; ~9 W2 X2 V+ c; `5 U6 o7 p( q
    C:=CyclotomicField(10);C;7 P) `: W0 [$ f( j2 _% u8 ^. i
    FF:=CyclotomicPolynomial(10);FF;0 X% C4 u2 E% C6 T

      H: o$ ?+ B/ }. v+ }0 [F := QuadraticField(10);
    . [; j9 }% h! t7 cF;3 M$ y5 @! o& m; o* C
    D:=Factorization(FF) ;D;
    5 X9 P2 Z5 {' j$ B' ^# @& C' xQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field# V$ r: U  |$ J5 O8 n2 ]6 E
    Cyclotomic Field of order 10 and degree 4
    " d& \1 q: o- l" @# ]$.1^4 - $.1^3 + $.1^2 - $.1 + 15 k; X, I! }" V5 T1 H
    Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
    4 M. m- l; H  f2 v& W; s+ {9 ~[
    , K. w" n: T0 W/ j    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>' h( f( b) y0 m
    ]
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