虚二次域例两（-5/50）

lilianjie

  N$ j4 ^# F% X8 ?. s3 A6 OQ5:=QuadraticField(-5) ;
Q5;
9 Q8 u& H; ^; E( S- t+ N2 U" m
& ~0 K: s" D5 G* U8 ?2 K1 e* T' a) \Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
NumberField(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;
IsQuadratic(Q5);
IsQuadratic(S1);
# S: P% H, ]9 O. j* k+ W9 `IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+5);  
; I" _6 t4 P# B2 ]$ K; Q+ c& g! s5 |Discriminant(Q5) ;
& n* W  w1 g" ~0 S1 P9 KFundamentalUnit(Q5) ;
FundamentalUnit(M);
) z7 w+ f5 ^# _% @8 {Conductor(Q5) ;

Name(M, -5);
. e% O; {% I0 H; y: M/ d: {Conductor(M);
6 F! Y6 H1 ]# U3 R( f7 {( D1 X7 MClassGroup(Q5) ;
: i3 L' r8 Z* n. c) ?5 sClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
0 x1 T- m4 P* vPicardGroup(M) ;
PicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
6 k# k" f1 \" O% X" [QuadraticClassGroupTwoPart(M);
3 A/ o' j  r+ b& k1 |1 k/ c9 CNormEquation(Q5, -5) ;
" V9 `3 B- H, z$ ?9 h- rNormEquation(M, -5) ;
$ O4 B6 {6 ~! a( m; bQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
3 X+ `. }7 ?8 u% Y8 tUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
& P8 P4 F8 I3 [+ tMaximal Equation Order of Q5
2 ~* m, I& G/ j# \" [: nQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
+ t6 J, \  x9 g6 n0 Mtrue Order of conductor 1 in Q5
$ D/ Q/ l/ \; x1 n  _# V) V* w: @true Order of conductor 1 in Q5
  p8 |/ m7 B0 U- etrue Order of conductor 1 in Q5
5 C0 x  v) \( v; o" E8 Y[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
% }) X( m9 u4 B  l# @4 ~! B# O]
/ v% V$ _4 R2 v6 q- O; t4 h; J-20

4 W3 J7 x  Z5 |( d/ Z% }$ q>> FundamentalUnit(Q5) ;
3 r* }0 F  m2 O+ m7 a' Z# r                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
( K1 T- L1 V( y! ?, I" D

( C: t6 w$ G1 h& A>> FundamentalUnit(M);
                  ^
! x2 R$ K1 c8 Y) B/ H  yRuntime error in 'FundamentalUnit': Field must have positive discriminant

* d+ n0 {. T: ^9 ?+ W3 v20

>> Name(M, -5);
       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
; c" l  ]4 \- z$ \/ RAbelian Group isomorphic to Z/2
Defined on 1 generator
$ {% C: n: U% m" `Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
" ?+ n1 c4 N- B  _; C, P    2*$.1 = 0 to Set of ideals of M
Abelian Group isomorphic to Z/2
+ g4 L6 ]3 ~* K# oDefined on 1 generator
Relations:
    2*$.1 = 0
4 R) g6 E- t7 h/ ^3 }Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
0 x. M; _9 i! g- m( IRelations:
) y& a/ U4 N* `6 |' S    2*$.1 = 0 to Set of ideals of M
  P; h. l' S* X/ b. t+ T! @2
2
3 b5 D- m; ~8 x; h! J) [5 RAbelian Group isomorphic to Z/2
0 W& C9 a( A2 L* `$ T9 `4 pDefined on 1 generator
( C' G+ w' ]/ Z% I) G$ J& c, _0 ?$ R" L4 VRelations:
3 x! `  L8 x# W1 W    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
; S- z# Q0 [) J. NDefined on 1 generator
2 d% q4 `" ~; b' W) WRelations:
+ D/ C! y4 ^. a' z# o) g    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
. \4 n8 P% y6 s1 a" O- P2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
0 X: D' W0 ?7 _3 aMapping from: Abelian Group isomorphic to Z/2
. E  x- `% ?6 z4 U8 qDefined on 1 generator
Relations:
1 Y6 y8 c6 _# S7 P5 `+ V  A    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
4 n3 D5 D% J2 d# [2 l# Q4 l/ ?* finverse]
0 `3 e4 @2 J$ g; t  d+ L6 @  TAbelian Group isomorphic to Z/2
Defined on 1 generator
: f! j9 u7 C, u2 p3 r4 TRelations:
6 w* I" F- `4 ?    2*$.1 = 0
5 P+ ^! E' i' ]9 w* nMapping from: Abelian Group isomorphic to Z/2
+ A. `+ t# q! B* }$ O5 \2 yDefined on 1 generator
' R9 d; z' n3 T! T6 cRelations:
  K9 Z$ x' f- Z3 I    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
' f' W# Z/ h' V7 r# p- Sfalse
false
==============

2 E7 @" Y) \) w7 g! ~2 ]. ~& R
2 q7 \  Z/ C$ n! y1 y% MQ5:=QuadraticField(-50) ;
& H# t/ J. @0 X. r- B2 s% V; [$ XQ5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
. {4 Y" X9 B( \# S- rM:=MaximalOrder(Q5) ;
M;
NumberField(M);
6 X& p8 g' D. A: p# z$ wS1:=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;
IsQuadratic(Q5);
9 N* F0 M, b$ t. n8 G8 gIsQuadratic(S1);
5 U# d2 b' \- JIsQuadratic(S4);
, h) ^: U. U+ j/ ]! J, JIsQuadratic(S25);
) {4 f2 e; @  L0 N( j! E: UIsQuadratic(S625888888);
Factorization(w^2+50);  
Discriminant(Q5) ;
! ~* \0 `$ M1 P! VFundamentalUnit(Q5) ;
9 H" x, A8 x! Q/ A* s1 LFundamentalUnit(M);
Conductor(Q5) ;

1 T- U7 ~  k+ I* ]Name(M, -50);
2 X) D  m2 m3 U1 Y- B  cConductor(M);
ClassGroup(Q5) ;
9 N1 Q  G. T, }' jClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
2 c) x8 w7 l# s& o( X8 LPicardGroup(M) ;
PicardNumber(M) ;
8 @) d7 w' _1 B: U$ X' A8 s* W1 \
. h3 |6 s5 V0 `( q3 p, Z6 uQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
. m  G3 y/ z1 K1 H: s7 FNormEquation(Q5, -50) ;
NormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
$ d- t: h0 G* M- }+ S6 JUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
1 q5 |- \/ m& Q4 {; M* J$ LQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Order of conductor 625888888 in Q5
  I$ Y; {6 _+ l8 W$ Strue Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
$ M3 J" k+ V: Y; U' etrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
+ t6 u& Q3 V5 Z( c6 ^5 R, p[
    <w - 5*Q5.1, 1>,
. E% k8 N; o0 |3 Y8 @" x. |    <w + 5*Q5.1, 1>
]
-8
- q& L' K) k) i; o
>> FundamentalUnit(Q5) ;
% j) T/ \$ b3 V- H0 e7 G2 l% A& e                  ^
/ p, G3 U# `/ g7 |7 W: J5 U  MRuntime error in 'FundamentalUnit': Field must have positive discriminant
9 G: @* g7 d* P$ O
7 J; z1 C  W& o8 {0 x/ D2 H
>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
& h: A/ L" M! M! N6 n4 |7 a
3 C5 t: T% v6 N( s% \# M+ j8

>> Name(M, -50);
1 j+ N: d/ k- F$ y3 f& S( R7 _  ?! B6 q       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
( q. @+ g+ u& u
1
: B7 T5 v, d- \5 A5 p1 Y! EAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
6 ]4 ]- g. Y( j1
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
, }8 O/ q% L1 p3 U2 r; I, iinverse]
: ?. j- B' R$ @$ K0 b5 j. ~! |1
: s% t9 z1 j$ R' h! _+ x$ |  XAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
. h$ A; U9 a% G- f( _# }6 n-8 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
, r. N$ D$ N! J2 U% i" bfalse
( E/ Q: B& @1 X& H2 bfalse

lilianjie1

分圆域：123

lilianjie

3 z4 h. Z# p* a' Y6 N2 @" i* b" G

lilianjie 发表于 2012-1-9 20:44
" U" l+ l% z  {8 b+ _* ?& O分圆域：
$ Y4 ]7 h  P/ NC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);

1 P0 l/ _) r. I( K4 T9 a分圆域：
分圆域：123

R.<x> = Q[]
F8 = factor(x^8 - 1)
F8

(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)(x - 1) * (x + 1) * (x^2 + 1) * (x^4 + 1)
9 T; n+ U) {4 ~( c: i0 `
Q<x> := QuadraticField(8);Q;
+ D& }7 ]9 T. q1 O% ^% z) cC:=CyclotomicField(8);C;
- V- T$ @7 w' w5 N# WFF:=CyclotomicPolynomial(8);FF;
) }; j5 ^  s4 }% F! e% }
F := QuadraticField(8);
F;
0 {: ?8 ^; y0 @8 m* M: S* t4 D, k/ f" LD:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
Cyclotomic Field of order 8 and degree 4
8 o$ p# h6 ^: h8 n* x) q$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
]

+ s7 y3 @# N2 ER.<x> = QQ[]
F6 = factor(x^6 - 1)
/ a. C1 _+ J6 }* D$ m. {7 }! X( vF6

(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)(x - 1) * (x + 1) * (x^2 - x + 1) * (x^2 + x + 1)

Q<x> := QuadraticField(6);Q;
C:=CyclotomicField(6);C;
% H. Q! Z! g+ \9 U* ?* eFF:=CyclotomicPolynomial(6);FF;

F := QuadraticField(6);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
: H. y3 U. G; L& T6 wCyclotomic Field of order 6 and degree 2
4 m9 I% ~7 A2 n$ @/ y; R8 I5 L5 b3 W$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
% y: A- k5 y2 o$ o  F  o) ^[
: N; H* I* }" D- k    <$.1^2 - $.1 + 1, 1>
]

* i6 E1 l& K6 ^, ~3 t$ f' DR.<x> = QQ[]
F5 = factor(x^10 - 1)
1 }/ J$ N3 D% @+ A1 bF5
9 ^' {  v/ Q/ X( _- a(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
. Z" C' f# T( B4 Z) l' U1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)
" @# R1 i: d  o8 \* `6 @/ W
Q<x> := QuadraticField(10);Q;
4 R( b* r  q* v% l( N* Z4 `C:=CyclotomicField(10);C;
$ m9 \# ]. H' X. @$ t$ EFF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
' a- e* W+ |; {( t& ~; x* H, nF;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
Cyclotomic Field of order 10 and degree 4
( M3 W' r, D% `5 x$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
( @+ G+ n/ s0 @6 {; b[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie

6 z5 j2 `$ J+ G) S


; V5 V3 U  m7 u  {/ X5 u
) E! M' G2 g6 f( z+ x/ U2 h


) L; D6 L9 O1 D; B分圆域：
C:=CyclotomicField(5);C;
$ a: G8 `0 D" YCyclotomicPolynomial(5);
C:=CyclotomicField(6);C;
1 ?2 w" ?9 g6 ?CyclotomicPolynomial(6);
! Y) c8 e2 n; j& r7 Q9 wCC:=CyclotomicField(7);CC;
* I) a" Y) ?7 a8 I3 UCyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
0 B, K( _  x. K0 d5 X7 a' mMinimalField(CC!9) ;
MinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
* U, L: t3 F9 V' JMinimise(CC!123);
* {0 f7 |" x* r0 m! \+ H) n- s9 ^$ |Conductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;
4 A+ [4 h  W, x
) n8 u/ b; h6 o  W0 L9 j/ ?! {5 iCyclotomic Field of order 5 and degree 4
2 w5 r- b& x. g  `$.1^4 + $.1^3 + $.1^2 + $.1 + 1
# ], N" S8 O' L3 h. p; nCyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
; _# t  w1 X, D: I) L+ `  q$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
% E9 K7 W, c& m& TRational Field
) M( f' {8 q! ~, p% K- i- rRational Field
Rational Field
; f2 K( x6 C4 |% G+ W6 ARational Field
zeta_11
/ B7 j* u7 Q+ j) x: Vzeta_111
123
: G- Y) d+ c# z+ Y" ^# D7
7
Permutation group acting on a set of cardinality 6
Order = 6 = 2 * 3
' ]- m4 m  m1 R" i    (1, 2)(3, 5)(4, 6)
" Y3 H/ [8 z# x8 K- Y/ C    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
/ M7 h9 o' C& v# ~% y# OCC
, e8 l  S3 f2 ~9 P: \" xComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
# u$ F% o8 C+ W1 ~% v; PDegree 6, Order 2 * 3 and
, e1 E+ g7 E  t, {! i" @' mMapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC

lilianjie

6 k/ c: I  C' c9 B. t
F := QuadraticField(NextPrime(5));

KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
8 N4 \8 [' c# D- }8 `' l9 S4 {Conductor(KK);
+ o- x: h7 u  G+ Z6 r3 m+ u; `8 z9 aClassGroup(KK) ;
QuadraticClassGroupTwoPart(KK) ;
  b1 j( g+ C* }$ w0 aNormEquation(F, 7);
* s8 M* h8 J! A7 xA:=K!7;A;
) _5 l, M% b/ P, ~  \6 oB:=K!14;B;
% A% ]- o7 C; S% cDiscriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
( J( [9 S) @9 ~$ J( v28
' o5 B2 D6 R+ JAbelian Group of order 1
/ W) S$ D$ z$ m. u. C2 q$ g8 [. H' wMapping from: Abelian Group of order 1 to Set of ideals of K
; R6 I& w8 g: w; S: f* x# qAbelian Group isomorphic to Z/2
& z2 w  e- x* ]- t" Q) {0 o+ E% kDefined on 1 generator
$ z8 }' O7 y6 h0 GRelations:
; k8 c" ?( M4 K7 s: A    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
/ U: U7 C7 J, s- Z* P9 t% UDefined on 1 generator
* m+ y" `% B" M# T# nRelations:
    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
) Z! s) a5 u, |7 g6 i  Kfalse
4 T6 C9 d# \* L4 k6 r" e7
14
- g$ O5 F7 s# a28

lilianjie1

pell   equation

lilianjie1

Dirichlet character

lilianjie

0 e5 q( K% M, R+ e' E
  Z9 j& X3 I( \- i" {4 w+ sDirichlet character
* g) A4 Y2 I9 @7 s' RDirichlet class number formula

9 Z7 b- q( r3 z3 f' [4 k; e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

, ~, c3 F+ p5 W3 o-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
' F. x9 G: n: C- Q/ v' y6 ~
0 U5 g2 v9 H& r( }* v; c-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
h=-6/（2*3）*Σ[1*1+（2*（-1）]=1
4 S4 j( D  T) g4 ^" X
-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，

& i- l6 T( S$ x7 ^- L2 U

h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
+ S; ^5 I+ G  d: c/ J" k
) x" [2 V1 q' m  s( [, K9 A4 u% }
. n6 \/ e! d* u) d
-50时  个单位根                          N=200
3 U) S8 }- n+ G1 H* }# ~6 ^' R7 u1 z

孤寂冷逍遥

lilianjie

看看-1.-3的两种：

/ M2 W3 j0 I5 }$ e% qQ5:=QuadraticField(-1) ;
Q5;

2 ~6 z8 ?- |+ B* j& M# Z7 oQ<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
; p) M- n. r4 l9 m3 \9 xNumberField(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;
IsQuadratic(Q5);
, R: j0 r. ^  [9 C5 S$ T' zIsQuadratic(S1);
1 ~2 _- t# h7 KIsQuadratic(S4);
IsQuadratic(S25);
  I1 p) w& x! KIsQuadratic(S625888888);
' i3 R& |. T8 m2 w# p3 r" `9 r  Q  @Factorization(w^2+1);  
* X; n. [0 p% l# ^+ K( s" vDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
" {; E" o7 l5 ?0 s' WConductor(Q5) ;

Name(M, -1);
6 v( G/ @9 B2 w' a& l# ?. }, dConductor(M);
ClassGroup(Q5) ;
- U& h/ K9 }$ g: ^! g. q2 J% kClassGroup(M);
/ g  x. s# k; v4 p( _: \% TClassNumber(Q5) ;
ClassNumber(M) ;
& X; d' g# C0 h' j: x/ kPicardGroup(M) ;
PicardNumber(M) ;

5 e2 t# m- G/ zQuadraticClassGroupTwoPart(Q5);
" B1 B- g5 g9 E: K! XQuadraticClassGroupTwoPart(M);
* R% j8 k" n  zNormEquation(Q5, -1) ;
NormEquation(M, -1) ;
( |/ n  L. u. }: \
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
7 F1 [% q% ^/ j5 v" Q( T* `2 VUnivariate Polynomial Ring in w over Q5
8 d8 u6 Z( B1 }4 fEquation Order of conductor 1 in Q5
Maximal Equation Order of Q5
: A9 f) l5 r+ b- L, [: GQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
# c2 `4 l$ @5 S2 e5 r" ]: t8 wOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
" N! g' r5 Y- d' Mtrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
! d1 C9 O# \6 p, d. _: rtrue Order of conductor 1 in Q5
8 K  f2 I  j$ ~$ C6 P1 S/ n[
/ p. U3 Z1 H7 S" I, D* ^    <w - Q5.1, 1>,
    <w + Q5.1, 1>
+ N! _+ S. A$ X9 |3 M- J]
-4
6 D2 u# X+ A4 G* r
8 Z: r& G& S* S>> FundamentalUnit(Q5) ;
! _7 W$ q) q) h0 T9 ]9 z1 v                  ^
4 O& u' x+ N% ]  M9 s% ARuntime error in 'FundamentalUnit': Field must have positive discriminant

  L$ n# B+ Q* E- ~! z3 `
+ U/ X8 h& ~/ p3 B$ g2 d) H>> FundamentalUnit(M);
5 f8 [- w+ @: i( M; W% w                  ^
2 I; V- q4 Z* {) K+ TRuntime error in 'FundamentalUnit': Field must have positive discriminant
+ ^/ B$ a7 X; |3 w+ [- F9 ?5 A* H
4

>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

' P- [+ `$ q. ]& I1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
1
0 j* m5 X; I5 H+ j$ I1
Abelian Group of order 1
2 O. j: f8 t  o5 z. b/ {* y1 AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
% Y' x& q+ x* A/ l2 [Abelian Group of order 1
3 z: p- k8 S, b. }  J" Q  ^" YMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
4 y) H5 _, p2 D9 x6 K+ |-4 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
false
9 D0 t& ?4 d0 }( A# l6 Y  C" F8 S1 Gfalse
  h7 B7 g8 W3 i, l) s/ E===============
& J' ~$ ~* y8 F
Q5:=QuadraticField(-3) ;
Q5;
1 k2 J3 [" X1 a, P
* ]4 q& A) p8 B, j: z9 b. m5 G9 bQ<w> :=PolynomialRing(Q5);Q;
/ T1 \6 ?% z* t  ]* r& iEquationOrder(Q5);
, b9 _8 D5 C' C/ R' xM:=MaximalOrder(Q5) ;
3 w' p% A4 m3 h' fM;
NumberField(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;
IsQuadratic(Q5);
* J& n" K& T6 UIsQuadratic(S1);
0 B" [6 N, G; [7 y8 GIsQuadratic(S4);
3 C& s: n7 H! J, p$ U- e, `2 zIsQuadratic(S25);
1 S( ~- ~" N4 X: i; t/ V" uIsQuadratic(S625888888);
Factorization(w^2+3);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
7 T$ v" q0 L  ^& ?+ R
5 S- r3 a% E; K4 E2 H2 t4 Z4 o* IName(M, -3);
- R+ R( K* T( B9 J! eConductor(M);
2 R1 Y& U9 A* pClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
  ]; A0 X7 `6 ~3 \PicardGroup(M) ;
PicardNumber(M) ;

! _7 d" w0 e% y- g3 T+ W1 s8 hQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
3 p3 E6 m5 u1 H/ N  j8 zNormEquation(M, -3) ;
  }% F* v4 K* l+ @, H! Y/ O! Q# r
+ o) D0 P" X. h! x  t; LQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
' H0 z& m6 K9 l  n  S" s! X5 JUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
: B: W5 i2 {* P* x4 M8 m& X1 mMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
7 x9 J/ k- u8 L; eOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
& x; o7 T; B, ~2 htrue Maximal Order of Q5
, O5 M) `( p' n) Q# ]true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
* M1 p2 b4 X* D$ W5 [0 i; Btrue Order of conductor 391736900121876544 in Q5
5 D. x2 v0 h" M[
& w, a1 p2 R2 T9 P- m3 t: l0 k1 ]    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-3

>> FundamentalUnit(Q5) ;
                  ^
; ?# `- u! q1 qRuntime error in 'FundamentalUnit': Field must have positive discriminant

! y& Q& S8 }1 f3 i, ?
>> FundamentalUnit(M);
$ j  q: B% ~  y. X+ y1 ^1 u                  ^
7 Z) i7 [/ |* w% B; H4 BRuntime error in 'FundamentalUnit': Field must have positive discriminant

8 ?- ^! N" p1 S! D( F1 M2 n3

9 }8 A* ^  x3 @3 F>> Name(M, -3);
       ^
0 S, S$ Y: e0 ~1 D- p  WRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
3 h- Q& b7 _0 @( p, y: |
- q9 o/ L$ V  {: K0 Q1
Abelian Group of order 1
; c1 p- ~  ~5 I- y$ B* gMapping from: Abelian Group of order 1 to Set of ideals of M
; e7 v" q0 J! ~; M  h* Q+ S" k1 SAbelian Group of order 1
3 W4 l+ u* A% f# q+ H+ c3 QMapping from: Abelian Group of order 1 to Set of ideals of M
1
" y$ b! }0 v& [$ A- L' R1
& u1 r9 `+ P! K4 c/ [' [. uAbelian Group of order 1
% y- l+ K/ _0 L. V: PMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
0 a' E) V, [+ p) W% \4 P% v1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
7 _. @$ o6 C" R- A, B8 f" o! [5 }Abelian Group of order 1
4 v6 h. k# C- u' z! jMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
false
# d- A" W# q% x8 D9 l3 Ofalse