* T7 }8 g( s- T, A( i3 gEquationOrder(Q5);- @4 R9 C$ Y) D& r, O; G h% O
M:=MaximalOrder(Q5) ; 8 c; A G) W9 O/ {! V! rM; : I3 P9 k7 a; ^, H, n, yNumberField(M);8 x) f8 q* M6 J, h0 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;4 Z1 i c6 @- b' J9 o$ O K+ N* M
IsQuadratic(Q5);IsQuadratic(S1);IsQuadratic(S4);IsQuadratic(S25);IsQuadratic(S625888888); 5 G$ z: i; `8 X/ Z, ?. H/ gFactorization(w^2-3);. {& s9 X( L* C1 b- D" D% k
Discriminant(Q5) ; 1 \- C3 T8 @* T2 [. J- ~FundamentalUnit(Q5) ; 4 A2 a# P6 ~ R9 {FundamentalUnit(M);( E$ }: G" U# z3 C& _# a; W
Conductor(Q5) ;1 H& S/ a, ]0 h" S: o4 m
Name(Q5, 1); 0 z) N2 N( d! m, E# j+ ?) AName(M, 1); ( H M$ @9 S% Q# k) O, q# XConductor(M);. V( E3 \, q' Z# a+ x
ClassGroup(Q5) ; : z4 M6 O& Q% E( r/ `ClassGroup(M); - M4 f8 J4 z' h3 v. C) PClassNumber(Q5) ; 5 j9 A3 [; m v, W% i$ o0 gClassNumber(M) ; - Y4 y H m# x! T% D. ]7 e4 q! {0 c/ I; e( z" j
PicardGroup(M) ; ; a! ~+ A/ z* _6 tPicardNumber(M) ; - d: l B T) u# {: ^# L3 T ( u+ Z# J0 g. |3 \* q% R, @. {7 s' X) @: v/ M# k4 H
QuadraticClassGroupTwoPart(Q5);4 U# e, G9 X: i
QuadraticClassGroupTwoPart(M); ! p6 E- t }) ^" ] * ^' f/ C( ~$ z1 X5 A; ^/ ]& C$ h8 m- r. h1 R- ^
NormEquation(Q5, 5) ;4 d' I1 r, e! V' \6 ?
NormEquation(M, 5) ;* B% M* `3 a0 F) G2 w
! T5 X" b4 A9 g8 M
4 V' b- q e8 {& w
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field7 J+ P$ A3 B# Y8 _! K
Univariate Polynomial Ring in w over Q52 A: `6 | l* l) `) z
Equation Order of conductor 2 in Q53 ]7 v7 S+ S" b) s
Maximal Order of Q58 ?2 \% } q, `+ N" K- n& O: E; S) Q
Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field4 C1 X* F& y/ i( ]/ M8 }# ]$ i3 o
Order of conductor 625888888 in Q5 ) X- x6 B" v1 ~$ V" Q8 Strue Quadratic Field with defining polynomial $.1^2 - 5 over the Rational Field " I2 Q* w1 t7 U/ s! j& s" @. _true Maximal Order of Q5 " B# r9 E& ]& Q# Vtrue Order of conductor 16 in Q5* I- X$ y% B) m3 i$ F* Z
true Order of conductor 625 in Q5 9 D" _! k" l \& Otrue Order of conductor 391736900121876544 in Q5 / ^' B4 y$ X( E4 U! Z1 D+ {[. R" R7 r3 v' t7 z3 b/ A( e0 q# k
<w^2 - 3, 1>) E: @2 J1 L( l9 @2 z/ a& z
]. h1 e6 F D* t* {3 Y1 W
59 m2 g1 j$ [) z6 K4 b: Z/ s
1/2*(-Q5.1 + 1)" F s0 w: m9 |) o& {
-$.2 + 1- W3 P% h2 [/ J8 g: T
5% p) v7 e. T/ S
Q5.1 ! X1 {, g6 f* F9 a, ~8 `$.2( ~! h& M# X) D; M2 G( t1 \# a
1 . q3 E Y4 n/ j1 @0 nAbelian Group of order 1, ?- R$ C) y8 q/ x5 b' T
Mapping from: Abelian Group of order 1 to Set of ideals of M ) l. R9 t3 |* \+ kAbelian Group of order 1. V' G3 H4 d$ e% q
Mapping from: Abelian Group of order 1 to Set of ideals of M% v/ n% F. C, e7 E( j$ N1 T* S8 v
1. z% `" Z( T# g, R# Q
1# q7 m G. ~* ^4 B
Abelian Group of order 1 ' q0 U; _1 F1 G n0 j7 ?+ ?& yMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no' B" e6 S- s1 x8 w5 f$ h
inverse], n- z, U; C' F% e4 B5 w1 L
1 * e' U- L) O* `3 ]1 H+ P v( x0 TAbelian Group of order 1- ^' L5 T( {) c9 I. D: f" q% w
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant3 d) J2 K, y8 x' W* d4 a
5 given by a rule [no inverse] ' {; P/ r( x( i+ bAbelian Group of order 1 " q7 D9 G& \6 LMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant/ h. \( `2 K- Z+ H
5 given by a rule [no inverse] ( [& X' F# G2 O- d6 n& ^# B7 ttrue [ 1/2*(Q5.1 + 5) ]* [( n7 ^% |' f6 W& g+ f
true [ -2*$.2 + 1 ] n; R9 c* H$ z$ O9 f. U' P+ r) z; j% J7 t% v7 N1 W( x. D
& c4 F5 }: R/ u3 x0 R( W7 ~1 \* Z( o
3 N0 D' C* L7 z8 d. e1 T
% L4 c- Q' Z$ ?: q9 r& k0 I ( ^% f1 I! m* Z. U. |. u1 {! ]; t6 z* y4 o, ~
/ ~1 q4 J" I2 [- g4 s5 W
+ @3 C: _3 ~, v
; M3 K& y% M. u) I# b
8 ?: Z" S' v8 r' L0 V- ?============== ; N9 ?- t! G6 D+ l0 k" I% a, Z( j 1 R! y3 X! i: S- {5 qQ5:=QuadraticField(50) ; ! Y& ^) Q- F/ P% QQ5;+ q6 ~/ o/ r# j* d( }0 X0 v
6 G0 V4 q$ R2 }( V! z
Q<w> :=PolynomialRing(Q5);Q;& o& ]6 z7 U7 [7 N( Z' y
EquationOrder(Q5); 1 T0 k+ K/ O& l4 c/ w& aM:=MaximalOrder(Q5) ; $ A) P+ m9 |$ a4 {; gM;- B: ]3 n/ e5 P) e
NumberField(M); 7 ^: M0 N1 O, {0 N0 Q: g+ P8 zS1:=sub< M | 1 >;S2:=sub< M | 2 >;S3:=sub< M | 3 >;S4:=sub< M | 4 >;S5:=sub< M | 5 >;S25:=sub< M | 25 >;S625888888:=sub< M | 625888888 >;S625888888;0 I) B d8 H) d) n3 B5 z: a
IsQuadratic(Q5);5 @) X+ |" G9 q% G! \9 \$ f
IsQuadratic(S1);) S! @! a' b6 C2 D0 a4 f+ H( n; V
IsQuadratic(S4); . f9 v5 T f. J1 YIsQuadratic(S25); 9 H- U3 L* k1 i4 BIsQuadratic(S625888888); 2 ^6 W; N# b. j/ b* r0 g7 hFactorization(w^2-50); . _' v& ^/ B0 }% w4 g3 l
Discriminant(Q5) ; ; P5 {8 z, z9 @8 A3 [FundamentalUnit(Q5) ; 4 y& s+ e! L, h' r$ _FundamentalUnit(M);1 `4 E8 ?- ~; A% ~0 c# H
Conductor(Q5) ; 8 O- z5 Q+ ^! J/ h: }* I. V# ^/ B2 a- |* p" a8 u N ]
Name(M, 50);% |! U: d' @, Q' f. e2 v, v% T' H
Conductor(M);1 k. W! h" a- Q
ClassGroup(Q5) ; ; a1 r" ?$ n1 {3 V, D
ClassGroup(M);( f; A5 [6 Z" l3 f6 `, n# u6 g
ClassNumber(Q5) ; 4 u; b7 V/ w- M; RClassNumber(M) ; 5 G$ T2 u* ^5 o3 j. X9 KPicardGroup(M) ; 4 L8 i8 ]4 J8 Y' `PicardNumber(M) ;% a) L( O2 V! c, y( c5 z. I
( _( u9 R! i7 D: c9 g w4 L# P
QuadraticClassGroupTwoPart(Q5);( O# \# |! O. {. p$ z" p! c
QuadraticClassGroupTwoPart(M); ) G1 K$ x1 Z! x6 `; ^NormEquation(Q5, 50) ;' m8 g' K$ u2 C& d3 f
NormEquation(M, 50) ; # I" B# M& m+ e8 }" e . v d* s5 j _. j) c- q+ H0 O$ QQuadratic Field with defining polynomial $.1^2 - 2 over the Rational Field % K" z8 H$ A' oUnivariate Polynomial Ring in w over Q53 N& S$ ]2 K7 }
Equation Order of conductor 1 in Q51 H* H2 n" X; E
Maximal Equation Order of Q55 c+ C! Z% H0 d5 X
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field I3 v/ ^3 F8 O1 NOrder of conductor 625888888 in Q59 S; e4 [9 X/ M+ v3 S7 k: u
true Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field 2 M% K+ o5 F. |true Maximal Equation Order of Q5 . |- e/ P0 Y0 G- Q: `true Order of conductor 1 in Q53 K; v- ~/ G$ q, j
true Order of conductor 1 in Q54 V2 L# z8 M' h. z
true Order of conductor 1 in Q5 + m. r* Z, @. C3 T6 `0 f3 Y[- N7 b* p9 O( B6 n8 A' I
<w - 5*Q5.1, 1>,- _9 A( y) _5 a/ z
<w + 5*Q5.1, 1> " o! x. q; \) ^9 |% L1 i]6 n8 N% e3 g- l
8) Y! z+ W) }7 Z4 R: @
Q5.1 + 1 * W J8 s+ I; @1 |& R/ H1 n$.2 + 1# E7 _9 E# v+ x3 U
8 2 J3 F! U" e Y) m1 c4 b+ O7 R& f6 K H$ q, M
>> Name(M, 50);4 ~0 ?9 q( Q0 a( z% Q# h0 g, J' C
^3 Y9 D& d, G' r* L+ f# e d/ t+ d
Runtime error in 'Name': Argument 2 (50) should be in the range [1 .. 1]8 ~0 s4 J; ]' ~% G- r% \
% X5 m( ?* A9 G, I# m& h
1 3 b1 A& p# t q8 s6 QAbelian Group of order 1 " y0 c H; Q/ s7 EMapping from: Abelian Group of order 1 to Set of ideals of M: m7 _5 J7 m2 l9 F: A: {" t
Abelian Group of order 16 ~, T! V) ^7 e# E, O
Mapping from: Abelian Group of order 1 to Set of ideals of M ' Z/ ` V8 q: b$ t+ G) |4 t" u1" |" a. m. O9 O5 m* c+ a
1 9 o' G# h- Q1 x( c6 yAbelian Group of order 1 . j6 i" F3 }/ E# O+ }Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no 1 ^2 H& Y" c7 u7 l+ k) hinverse] . d/ k# }- m: c' @" q- r1* Z1 m( R7 {" r7 W: K, _
Abelian Group of order 1 # e6 O$ H. r" s5 b zMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant! S7 B: I- p0 H9 ] Q% [0 ~ M
8 given by a rule [no inverse] 0 z, J( X3 M2 u8 ~% eAbelian Group of order 1 6 A V4 p9 ~. ] _Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant- ~- x$ L- Y- g3 M6 M+ s
8 given by a rule [no inverse] % k: F* X' S7 ttrue [ 5*Q5.1 + 10 ] & h+ t3 {+ J# c5 l, B1 ztrue [ -5*$.2 ]
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发表于 2012-1-4 14:05
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