虚二次域例两（-5/50）

lilianjie

0 r2 y5 M5 _9 ~7 E5 O& }5 fQ5:=QuadraticField(-5) ;
Q5;

Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
( m2 j, C& k! Y: K0 R% [/ W9 \M:=MaximalOrder(Q5) ;
# E' \! Y+ _. t0 _8 GM;
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);
6 n7 ~* w$ j6 ~) k2 }# mIsQuadratic(S1);
. k; W. b  m) c$ oIsQuadratic(S4);
3 G' K% u% S5 t9 F' w# e, hIsQuadratic(S25);
0 C, `! E3 ]3 k. p& S! j1 e$ jIsQuadratic(S625888888);
, G1 w# B/ N" y% W8 r' P7 z9 \* mFactorization(w^2+5);  
Discriminant(Q5) ;
6 V( c) K$ a( r+ R! b/ E5 EFundamentalUnit(Q5) ;
, w8 q' {/ v& }- l0 W! qFundamentalUnit(M);
Conductor(Q5) ;

Name(M, -5);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
3 ~( h1 M+ F* f8 i( S% pClassNumber(Q5) ;
& T# j6 u1 _& `! h9 aClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
( s5 g* Y- h6 Y( C8 P  E* m" P
4 p7 D. C  [0 [* l* rQuadraticClassGroupTwoPart(Q5);
: q+ V8 A5 s$ ]) i* qQuadraticClassGroupTwoPart(M);
9 L' \* ]$ L% J2 b6 I6 b8 z( ^8 UNormEquation(Q5, -5) ;
NormEquation(M, -5) ;
8 [) u& h- o' c# O' `: x: zQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
" K! u% I, q+ p* a" K7 GEquation Order of conductor 1 in Q5
, q; n( m2 g& c. {' L8 W7 u0 ~Maximal Equation Order of Q5
5 h0 }. N" o+ P% j: LQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Order of conductor 625888888 in Q5
7 m8 X8 H" N; S( D6 z/ ftrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
5 J% }# q' |; _0 Y) M' i  ~  E[
    <w - Q5.1, 1>,
6 E; h: |& R) `, L    <w + Q5.1, 1>
]
; b' T& j* E5 o7 M& w-20
; U  t# T' O- G) F
2 x: i# q2 p, \2 I7 j( x>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


0 ?- y! ]5 g# @7 W- P- _. M>> FundamentalUnit(M);
& o8 ~+ U3 @! m& R% U2 R                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
. ^! B  G. r9 M( g, U
20

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

1
Abelian Group isomorphic to Z/2
6 j, V! T4 ^7 D% ADefined on 1 generator
Relations:
( w$ F) {8 v- q( r& Y4 }# C    2*$.1 = 0
' K! x$ b$ d8 z* c. O# I. kMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M
  x) f; b! q* C' j! J* r! jAbelian Group isomorphic to Z/2
6 ^. J3 G4 t+ \# a" v) G2 l: A" qDefined on 1 generator
- [8 n, B* l# Q$ G8 J6 YRelations:
2 I, m) j# o9 Y# C+ J  ^' {5 r    2*$.1 = 0
8 {% J1 y% C+ a' X- n0 XMapping from: Abelian Group isomorphic to Z/2
6 a- m& T* g( [6 V9 MDefined on 1 generator
! ~" z& n0 K4 T" ZRelations:
) r" O4 ~( _8 \- F    2*$.1 = 0 to Set of ideals of M
- C$ Z% u9 e: |2 Q3 _1 T& y# h2
1 r4 g7 \4 U8 ?2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
. C6 |, _( b- q0 b4 d, T& YDefined on 1 generator
Relations:
; p) H, f& |3 @- L    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
Abelian Group isomorphic to Z/2
3 F. A3 k2 ?- }3 S& hDefined on 1 generator
Relations:
0 K* P! b7 e% t# A* n    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
# O) C+ d! a' b) S" w    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
false
1 \( Y' b0 ?7 A8 _2 c, s& B! i: xfalse
==============


/ v4 t, ?3 ~, R$ h2 aQ5:=QuadraticField(-50) ;
Q5;
; R7 U* t) f! q' \$ A2 G6 M
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
1 r& F! B0 j. ?& t+ H* ?& k" NNumberField(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;
$ A( O5 Y7 r% ~9 o% N1 dIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
$ G3 L, b; p2 I( f' ]IsQuadratic(S25);
IsQuadratic(S625888888);
Factorization(w^2+50);  
9 `) b; ?4 C5 x6 j) }Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
* e6 M6 G0 z' O( EConductor(Q5) ;
1 x3 Z7 k2 c9 ?
Name(M, -50);
! x( b& Y, O& w- ]6 K7 ~Conductor(M);
% ^7 K) h5 A5 E% wClassGroup(Q5) ;
ClassGroup(M);
' ^$ `9 F5 ?1 e/ H1 MClassNumber(Q5) ;
ClassNumber(M) ;
: [0 j3 W" S# _! \: Z, B. pPicardGroup(M) ;
- L6 C) x( k/ o$ lPicardNumber(M) ;

, J- R& d5 ]8 ~4 j$ DQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
  T, F5 V7 Q! y6 N4 b3 KNormEquation(Q5, -50) ;
8 E, ]( U3 a( t% HNormEquation(M, -50) ;

Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
5 N$ Y, F6 |% U& w1 i8 _Equation Order of conductor 1 in Q5
& r: P! q/ H* X% Z3 nMaximal Equation Order of Q5
& b: `! g" P  d4 Y# TQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
* |8 Z: D, ~; d; J: T3 lOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
8 U% {% ?/ M% vtrue Maximal Equation Order of Q5
3 V+ w3 n( O. F6 x) [- `2 strue Order of conductor 1 in Q5
/ p! N! }2 ^( R2 W' G& q2 [9 ytrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
[
    <w - 5*Q5.1, 1>,
8 T5 a! ?+ T$ S$ M$ d7 [  E. t- e    <w + 5*Q5.1, 1>
]
-8

; r/ B: l! ]  K  A; h>> FundamentalUnit(Q5) ;
# A1 y2 J0 y% ?. F# q                  ^
5 }1 d3 Z4 Q4 N' |0 |' ARuntime error in 'FundamentalUnit': Field must have positive discriminant
0 o2 a+ J1 y" |. Y/ N- t9 v' _

>> FundamentalUnit(M);
, ]4 h: b) s2 R  p                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

; @4 W: w. L) G8
! u) k& \) U& A. }9 n
5 N- B: B! r# V8 p, \>> Name(M, -50);
* p; l: Q/ b7 F6 y       ^
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]

1
5 ?1 z2 F$ q1 `: B3 HAbelian Group of order 1
# ?6 t9 w4 B  `Mapping from: Abelian Group of order 1 to Set of ideals of M
2 t, o7 i6 m9 ^2 g) jAbelian Group of order 1
) C7 c6 F3 O: YMapping from: Abelian Group of order 1 to Set of ideals of M
1
, t# m# F. K9 u, r8 l1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
0 ]: l1 \2 o% n9 D* ~* M1
1 @# L: a$ Y/ k4 _" S% KAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-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]
( P7 ?$ j7 J  C1 c+ U! q6 zfalse
false

lilianjie1

分圆域：123

lilianjie

" A& p2 y8 E- w4 s; Z

lilianjie 发表于 2012-1-9 20:44
7 `( `2 v" X8 A分圆域：
) L* d) V2 n/ ~C:=CyclotomicField(5);C;
0 c0 ]$ j# P8 D5 Q0 iCyclotomicPolynomial(5);

% U  j1 k" q; B( z1 _% u. h( e# D# f
5 J/ R+ L, F7 Z/ B0 v分圆域：
分圆域：123
% f3 D5 p, t. y" o: |- `: k) z
( P5 I; n# x& t# a* i; ^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)
5 G& Z" A1 q% }, |
Q<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
FF:=CyclotomicPolynomial(8);FF;

( ?) q! P9 r4 Y, U: TF := QuadraticField(8);
7 C6 O; X6 I  D: }F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
+ x: q4 N, @9 ?  I' }  L. YCyclotomic Field of order 8 and degree 4
  y% t' `* [- r' @' Q! E, J' a) k$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
: i' O. C" [" Q8 j    <$.1^4 + 1, 1>
]
8 q: a2 a5 H3 Q' a# f  M9 g' e
/ }1 L& k  p* s$ {+ nR.<x> = QQ[]
( \- B9 w/ ^. f- }1 Y' KF6 = factor(x^6 - 1)
& L$ X+ F$ t6 M9 WF6

' K  e; z& i4 {  K6 b; O, Y(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;
FF:=CyclotomicPolynomial(6);FF;
3 O4 ]: e+ k4 {
& s( }, R- d% N' a8 X9 ^/ v/ uF := QuadraticField(6);
8 o$ {0 G! b# fF;
" }5 G) W) D, |" c! `D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
' [8 {0 e8 t/ o6 |; b: w[
4 B: h6 m9 k: ?3 M( T! D    <$.1^2 - $.1 + 1, 1>
]

( p0 t2 f3 O0 UR.<x> = QQ[]
; @0 ^/ e+ M7 y; _; x+ y- {$ ?3 lF5 = factor(x^10 - 1)
F5
  U, F8 B7 c8 M0 D) a(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

4 o" q  @* _: p" p6 R) {Q<x> := QuadraticField(10);Q;
C:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

F := QuadraticField(10);
F;
# u4 S4 r; m/ s( q/ LD:=Factorization(FF) ;D;
1 o% h; o+ C9 c- I8 M, `+ KQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
! y+ v7 T, x' R# Q' VCyclotomic Field of order 10 and degree 4
) G5 H" H8 l0 j9 M6 {7 O3 }$.1^4 - $.1^3 + $.1^2 - $.1 + 1
Quadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
( m6 ^7 y8 @; E5 I! v[
6 b2 ~1 l" k; E7 D& A" S, v    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie

; s  f* F& e: \

! B  q+ ]9 ~% t7 P3 `9 E
- L- ~" Z- E3 r. ~5 c0 M, s' i分圆域：
0 z( y# j$ b- }, ?- y/ n+ mC:=CyclotomicField(5);C;
CyclotomicPolynomial(5);
2 X0 \% u) A. j) V8 z; _/ |/ y4 bC:=CyclotomicField(6);C;
CyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
CyclotomicPolynomial(7);
MinimalField(CC!7) ;
MinimalField(CC!8) ;
" M" z4 K$ l1 C6 m/ IMinimalField(CC!9) ;
/ y4 {: R& D- p" Q/ Y  R' O; UMinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
! I/ h# S  @- e  @Minimise(CC!123);
0 s7 }; o5 F1 T0 MConductor(CC) ;
& G: {; I1 _: y4 QCyclotomicOrder(CC) ;
* ~# @" T9 Z3 Z( x# @1 d' |
CyclotomicAutomorphismGroup(CC) ;
: H2 o& n. O5 @/ m
Cyclotomic Field of order 5 and degree 4
  ]( t# p$ E% x9 T7 |$.1^4 + $.1^3 + $.1^2 + $.1 + 1
0 I0 m# g+ f+ y& |+ Y, A4 r0 N# I" GCyclotomic Field of order 6 and degree 2
6 a& k1 c8 s/ Y$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
5 i8 ?3 z: t9 @& R. I" h$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
" V0 O3 @% U* e! `! @Rational Field
7 c! n9 r! `$ X/ cRational Field
! L8 x, Q8 \/ h+ h* _% HRational Field
+ l" j, B/ M+ {' FRational Field
zeta_11
zeta_111
123
$ }, ~$ c, f3 ?. K; c4 X; U7
( I& V( c! `+ t! p" u, U' C) v* y7
+ ~0 C, K& J- mPermutation group acting on a set of cardinality 6
  U' O) J6 z$ r* Z" q) G1 ROrder = 6 = 2 * 3
2 u) }3 O4 I) n' F6 F- }. V    (1, 2)(3, 5)(4, 6)
4 C; _+ l4 r4 S% [: O" A2 M    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
) K, o" m3 L- y# L. PCC
  S' U! B+ _0 P5 F0 Q  l1 ]( wComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
Degree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
1 `$ n, k& P: U4 uCC

lilianjie

7 L% n5 @+ u4 v8 h5 I0 [6 O9 v/ g
F := QuadraticField(NextPrime(5));

/ R# U- k" ^& ^KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
Conductor(KK);
* s( s% Z' X0 J1 B; @1 rClassGroup(KK) ;
, X2 X# E* k& C1 s8 J! qQuadraticClassGroupTwoPart(KK) ;
: d' F4 z! Q. R, H$ p( X+ DNormEquation(F, 7);
! L9 L+ \7 \9 P5 t) C. uA:=K!7;A;
6 v3 G- O% q0 A$ b+ N9 x) n% F: UB:=K!14;B;
Discriminant(KK)

Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
# q$ J) q1 d8 h# C. l% N* Z- P28
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of K
7 v% c0 S% e6 n9 M4 JAbelian Group isomorphic to Z/2
Defined on 1 generator
4 y# `3 _- C1 H' |) b  Z% `4 GRelations:
" Z! ?9 O% {, f( ]2 H9 S. ^$ H" I    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
5 F8 t8 P) p: Q$ jDefined on 1 generator
! U) O" B$ V% N) G5 ZRelations:
6 v) S9 g- C) \    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
. [% D2 n% J+ T- X* Yinverse]
5 B0 R' d2 s% f! U/ m9 Nfalse
7
; X9 g& E1 i# k! V14
28

lilianjie1

pell   equation

lilianjie1

Dirichlet character

lilianjie

2 s6 L6 Q  B: I% }( B' X
$ e: M( F$ D' x2 U. D7 K& tDirichlet character
6 y- h6 P( {3 Q* |Dirichlet class number formula
. B5 n1 x1 Z  h, s7 [6 j* a( V
! z4 g* i, n* C. |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
# a1 {0 @$ K, F& ]
$ q% d3 d' |& Z+ g) V-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

: T- {7 I0 D- s; G. s/ L-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

' R* R! T7 p: V1 h' {" s3 z-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 O& K  D* ?6 U# s, F: q; c
; O+ Y/ l# ?3 F9 |+ t& B  x
2 T2 J1 q$ Q' _' Ah=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
- f9 n* X, L5 K# P1 \% a
% y! k3 b- V+ P3 m; ^. J
( |' s! s5 r4 G7 ]2 \3 Z
-50时  个单位根                          N=200

孤寂冷逍遥

lilianjie

看看-1.-3的两种：
+ B! ?- M2 [1 k; U
- R0 Z9 Q+ N) I! ]: z8 J; w. BQ5:=QuadraticField(-1) ;
( x# G, J& d* R5 s6 x$ OQ5;
$ E/ u5 r8 |  f; C& p
Q<w> :=PolynomialRing(Q5);Q;
" W' o5 N/ n/ T$ q- _2 c) |) UEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
( \3 c* n* w% t) T* Q3 G' B. S5 gNumberField(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;
- C; k0 |/ @' T8 e1 a9 N' O' }IsQuadratic(Q5);
IsQuadratic(S1);
, r  S2 @" x0 H' MIsQuadratic(S4);
IsQuadratic(S25);
, P7 B8 a* C# @( m: \. {1 @3 \IsQuadratic(S625888888);
Factorization(w^2+1);  
Discriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
7 K7 L) d9 j4 _; S- e
5 B% V% I2 }* F7 t% AName(M, -1);
/ l; T- P% B: R. R  S$ \Conductor(M);
$ i0 P$ i6 E  S, m5 m) A1 XClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
! S' i. m/ H4 j. DPicardGroup(M) ;
PicardNumber(M) ;

2 h6 a% T: S' `; DQuadraticClassGroupTwoPart(Q5);
! _/ }7 W0 _$ ^QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
: q) T4 {6 E! w) ~( `- tNormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
% G" i4 p0 E0 J* ]  IUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
; g, X1 v$ G2 Q+ {. |- l* SQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
2 w1 f9 ], z7 I% M+ t0 W) Ytrue Maximal Equation Order of Q5
true Order of conductor 1 in Q5
, ]! v& D0 X2 E" ^true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
' Q' j$ m- F  R/ f" k% a[
/ Z3 T6 b& \- c: X  Q" i    <w - Q5.1, 1>,
2 Y! S$ J  e0 e    <w + Q5.1, 1>
/ |( R$ q7 x& n' W& d+ D5 J]
-4

5 D( ?2 [+ J6 S# H8 U>> FundamentalUnit(Q5) ;
- h9 |  v+ \0 E# k, j2 B3 M) `                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant


>> FundamentalUnit(M);
3 y  C. g3 s$ R4 I% b                  ^
% k# ]6 `# H1 T" H% `# f. @Runtime error in 'FundamentalUnit': Field must have positive discriminant
3 H6 |. j7 f5 ]$ m* a/ \
$ P0 ?7 L  O. K8 X6 S- y4
+ Y7 |4 z& ~8 [% w' N# x9 U
+ Y2 \, x3 g6 l2 o1 I>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]

) N7 \; J5 M2 V/ Z+ p1
7 i+ `  B0 Q/ y# I# U+ lAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
4 I0 h+ e  q& B( X! H# h' Y0 QAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
; f, V- U! x& Q7 p1
! ?& P. R' |: A1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
9 P4 _$ w$ X" R6 G2 v1
7 z* @9 Z! {5 l, N) N# NAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
6 M+ r8 L3 {; @4 h/ y-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]
# a# ~2 H4 q% A! sfalse
false
===============
8 m$ C9 Y: e, k
Q5:=QuadraticField(-3) ;
Q5;
+ p/ Z% v( i. F: R' ]; S
Q<w> :=PolynomialRing(Q5);Q;
EquationOrder(Q5);
, Q7 k& ?' J& R" }4 C' E( {% E- C7 F  NM:=MaximalOrder(Q5) ;
, a1 ?* ?$ J. K" XM;
9 x6 X2 J8 P0 G  R8 C/ [* Q7 INumberField(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;
0 R) `6 R* V/ d' c- rIsQuadratic(Q5);
  f6 U5 V/ e: ?( W! V* pIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
IsQuadratic(S625888888);
9 v0 Q% z) n. e# {7 c1 S; R2 l& xFactorization(w^2+3);  
+ q8 G# N1 @* l: i  u( Q+ lDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
$ q3 r- L- e2 GConductor(Q5) ;
% g& N4 J# Z& _4 p- L2 }
Name(M, -3);
Conductor(M);
ClassGroup(Q5) ;
) o* ]; X% V7 f: [; bClassGroup(M);
$ V5 H: i) S( D1 S; GClassNumber(Q5) ;
- e5 _. b$ X& m' a, F8 w+ SClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;
/ Z. [3 s8 D4 _1 j8 X; e& h) }
QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
& ?' Y3 ^- i2 o: f' {# M( K' FNormEquation(Q5, -3) ;
NormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Univariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
0 p: H5 j/ y8 Y9 W/ NMaximal Order of Q5
. g$ F% r6 {. HQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
5 {2 d! A+ W4 n5 C, N$ p$ f7 XOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
true Maximal Order of Q5
3 ]) q) P1 y  u- F" ptrue Order of conductor 16 in Q5
true Order of conductor 625 in Q5
$ ?& |" f# ?+ t0 Q, Dtrue Order of conductor 391736900121876544 in Q5
5 u" N% T3 a2 P0 E) v4 G$ V, c5 t[
    <w - Q5.1, 1>,
    <w + Q5.1, 1>
]
-3

% l( S) s% }/ K+ j* O; {4 q; r>> FundamentalUnit(Q5) ;
3 p, l& n+ P0 [5 d4 D1 z                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

' @& ^+ q% l+ g! W  h- c3 B
& J+ l+ M+ q* A# g5 ?0 e, h4 y>> FundamentalUnit(M);
                  ^
0 T* F! x( |) ~. ^; A/ S9 D/ @2 l; ORuntime error in 'FundamentalUnit': Field must have positive discriminant

3
# H$ [& N4 N; a
>> Name(M, -3);
       ^
Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]

1
Abelian Group of order 1
! X" t8 y& g) S1 d" A4 p$ M! L- TMapping from: Abelian Group of order 1 to Set of ideals of M
) {/ D. F" I* p5 O( n( B( g; y! V# jAbelian Group of order 1
9 P* M  ]. }' D( }0 p! s# JMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
Abelian Group of order 1
0 z' h$ [/ h/ JMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
0 [1 _; y6 f5 J- L. a* U. ~4 Yinverse]
1
9 m3 y$ r9 z$ ?# l" m8 IAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-3 given by a rule [no inverse]
5 j' m2 f. b  a! v( z9 H/ }Abelian Group of order 1
" P0 {/ z3 g' W1 w1 |+ RMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
; r+ ]4 R8 a, L. i6 m-3 given by a rule [no inverse]
false
3 `( Y7 a" P! J' afalse