虚二次域例两（-5/50）

lilianjie

Q5:=QuadraticField(-5) ;
Q5;
9 g3 t7 {0 r/ J) B$ s$ }. k0 ~6 d
1 c( C0 S* ]& P0 h3 W, kQ<w> :=PolynomialRing(Q5);Q;
7 X5 D/ W% j  z2 w% s( C! KEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
  J( ~( A" N( 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;
) r* w0 n/ o' ~, K. CIsQuadratic(Q5);
IsQuadratic(S1);
IsQuadratic(S4);
! T4 `& Q. a! }3 _1 WIsQuadratic(S25);
7 R( d3 E- t* v/ D. rIsQuadratic(S625888888);
' ^* P+ y- G9 Y" DFactorization(w^2+5);  
Discriminant(Q5) ;
5 @0 T% J; {/ u! U2 g; b0 |FundamentalUnit(Q5) ;
9 n; r+ Q7 K5 L% z- s* YFundamentalUnit(M);
Conductor(Q5) ;
$ V* |: s- c% g% y6 P
Name(M, -5);
Conductor(M);
ClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
. R1 E( H5 ?* ^0 i# |9 \ClassNumber(M) ;
9 L& p$ u* R3 O  d6 r1 nPicardGroup(M) ;
PicardNumber(M) ;
- S7 t* m  i$ h
! G, V2 t, K+ h( bQuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -5) ;
7 q: X0 T* i1 o4 Z( c, c, _NormEquation(M, -5) ;
/ i" ^. y+ e  N' [Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
Univariate Polynomial Ring in w over Q5
3 D8 R/ l& [7 T) a0 N) v7 fEquation Order of conductor 1 in Q5
* I' |/ |1 _: A/ i; d- ^* qMaximal Equation Order of Q5
5 t  S" C. ~4 V3 f1 V9 MQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
& d* [2 b1 G( LOrder of conductor 625888888 in Q5
0 _2 \. K% R& R& {$ N0 ztrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field
4 T' U0 E8 d+ n6 |) S  R' c+ ltrue Maximal Equation Order of Q5
# G7 v" U( g0 C' X" g! U5 Mtrue Order of conductor 1 in Q5
true Order of conductor 1 in Q5
/ A. u- J7 f% [( y/ Btrue Order of conductor 1 in Q5
[
2 a! `% ?& l- E3 Q1 i3 j    <w - Q5.1, 1>,
    <w + Q5.1, 1>
0 F' v& G) Z9 E9 Z4 ~; J- i! u8 {]
-20
. }1 D3 R5 P& ]( q+ K! h% W
>> FundamentalUnit(Q5) ;
* K5 [0 y5 K$ h% t. i9 h                  ^
! p- ^4 k; E9 @: m; c; TRuntime error in 'FundamentalUnit': Field must have positive discriminant
# _, ~+ _% _8 r0 }. G8 g

>> FundamentalUnit(M);
                  ^
1 |+ z$ [5 p" \2 Q: v, gRuntime error in 'FundamentalUnit': Field must have positive discriminant
/ X* E+ h7 ?& z$ J2 p
! r5 ~2 H7 a" ?2 T$ F9 S. x# s20
9 L1 h) h# m) C! V: l$ E# J
>> Name(M, -5);
       ^
Runtime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1]

1
Abelian Group isomorphic to Z/2
Defined on 1 generator
0 ~* q2 z* |& \$ ARelations:
4 j5 A& ]2 F5 `: \) S: y/ d    2*$.1 = 0
* x) G" c; e& l1 SMapping from: Abelian Group isomorphic to Z/2
, z- \- r. }6 J0 l- \. CDefined on 1 generator
' S) U; P- @1 q; I  }8 JRelations:
    2*$.1 = 0 to Set of ideals of M
9 r1 H" S4 l7 K' O; z1 m% _' |  w: oAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
. B3 q6 a; j: b4 T4 h# F3 x6 OMapping from: Abelian Group isomorphic to Z/2
2 ?! i, V9 i% Y6 E& k- lDefined on 1 generator
) P% \- S6 _3 ?. v! H6 P1 s& K* tRelations:
    2*$.1 = 0 to Set of ideals of M
2
2
: W( l& H# i% p/ F/ O5 V. g) rAbelian Group isomorphic to Z/2
  [* [' K0 H9 n0 S7 x4 N' HDefined on 1 generator
Relations:
5 f6 U( G( t6 o, {3 P    2*$.1 = 0
" R8 `7 ?% W  |/ m( }! wMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
  D: g8 E4 \4 R. z: o9 q# B  _, @; DRelations:
! J/ V6 ?* X; k2 R' O0 X+ O    2*$.1 = 0 to Set of ideals of M given by a rule [no inverse]
2
Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
Mapping from: Abelian Group isomorphic to Z/2
, |8 c& ?% u9 K: b, MDefined on 1 generator
Relations:
    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
inverse]
3 q4 F8 ]3 u' C# TAbelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
    2*$.1 = 0
0 d6 q% E# U; H+ @4 ]Mapping from: Abelian Group isomorphic to Z/2
( h& }, B8 N  A" gDefined on 1 generator
) i  y2 j3 U! _Relations:
$ R& n4 h, T8 ]7 D% j    2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no
8 V, i& ^! Z+ `$ O/ Finverse]
# k8 D! C0 b8 _" K( Sfalse
false
==============


Q5:=QuadraticField(-50) ;
! Z- m, Z! j: j5 PQ5;

Q<w> :=PolynomialRing(Q5);Q;
- n3 D3 o" y% q: r& REquationOrder(Q5);
M:=MaximalOrder(Q5) ;
) Q# C( z5 A( ]9 Y+ EM;
( k! _1 J( x/ `" a$ vNumberField(M);
) I4 [, b; E6 A3 K6 M2 W  i8 NS1:=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);
& _, ?/ D  ?- f' Z& q, f! i" k0 pIsQuadratic(S1);
IsQuadratic(S4);
IsQuadratic(S25);
4 h& Q/ K. {! p( O5 DIsQuadratic(S625888888);
# M1 G! w5 J2 A8 z; v! C% IFactorization(w^2+50);  
- W: i& f3 Z. |0 \Discriminant(Q5) ;
FundamentalUnit(Q5) ;
7 R9 W" M: q$ Y- b! r( UFundamentalUnit(M);
Conductor(Q5) ;
* S- _8 Z5 v" r) j3 ^
Name(M, -50);
Conductor(M);
' D/ A* n9 Z  [) R& x1 O9 ]ClassGroup(Q5) ;
. U6 S9 f0 r6 @1 d4 k3 o) WClassGroup(M);
' P: w4 q1 ]) Z3 q8 \) MClassNumber(Q5) ;
3 [5 |& Z( d- W2 F: [4 W0 |ClassNumber(M) ;
PicardGroup(M) ;
PicardNumber(M) ;

3 u1 w8 w( D0 j1 Y- bQuadraticClassGroupTwoPart(Q5);
; X" t9 q) D0 ~QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -50) ;
3 Y0 {, g9 M0 u/ \NormEquation(M, -50) ;
- V, o6 ~! [3 I9 {" d9 @, _
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
Univariate Polynomial Ring in w over Q5
1 R/ N3 C: p6 o" `Equation Order of conductor 1 in Q5
  B2 z! G2 ]9 a3 P4 f7 _, L) l9 y0 e2 QMaximal Equation Order of Q5
7 g6 [: l3 }% \Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
: D: L: m1 K2 a: V4 R& I3 W0 GOrder of conductor 625888888 in Q5
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
- ?) L5 K# f9 f, q/ c2 f6 Ktrue Order of conductor 1 in Q5
[
1 c4 l- |0 t. }+ H+ N$ X    <w - 5*Q5.1, 1>,
7 I/ o. z$ E9 D! U5 Z! x    <w + 5*Q5.1, 1>
6 ?. E) K5 W7 o+ K1 T]
-8

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
6 d2 h0 y7 n3 `2 R9 ?+ W" C
: V; [0 B& R% m* x: l; h: r
>> FundamentalUnit(M);
                  ^
0 j) a& ^3 o0 k. W1 r  R2 [5 q3 iRuntime error in 'FundamentalUnit': Field must have positive discriminant

8

>> Name(M, -50);
       ^
$ F' m) v& r1 I. E, ?- @Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]
! _/ {( I8 u  s) I8 w1 j- h
1
7 x* W, T* u/ X) y, oAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
3 g9 [) f# A0 V7 `# R  BAbelian Group of order 1
. Z  q  v2 T9 f# a# x* gMapping from: Abelian Group of order 1 to Set of ideals of M
1
1
: C; U" R1 `1 o  z( I% ]& rAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
" G( p, o# @& u7 d2 i1
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]
* g3 }& N9 z# i7 z- V" WAbelian Group of order 1
) u1 p3 {' G- I4 z# ?: W, v( f( o0 F7 qMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-8 given by a rule [no inverse]
! e; q# B6 y2 C- c' D0 afalse
false

lilianjie1

分圆域：123

lilianjie

# _; ]/ s$ s( S2 S  d1 E+ U

lilianjie 发表于 2012-1-9 20:44
/ z+ P# u) h' I* F# _3 {/ n' O分圆域：
- G: {# v8 ?9 PC:=CyclotomicField(5);C;
- S8 W8 q7 l1 C- D5 ?CyclotomicPolynomial(5);

分圆域：
分圆域：123
+ w3 {: W( Q" ^( `
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)
' R& ^  b" M9 V6 ]2 ?( J+ z
! W1 o5 W, B0 h: Y# Y% R1 F0 r7 G9 XQ<x> := QuadraticField(8);Q;
C:=CyclotomicField(8);C;
' W( t, I& q  c7 d: }FF:=CyclotomicPolynomial(8);FF;

F := QuadraticField(8);
F;
D:=Factorization(FF) ;D;
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
0 `, L# _1 C  p2 Q9 Y5 k' E2 NCyclotomic Field of order 8 and degree 4
6 |8 L0 V% m, D- ]$ x$.1^4 + 1
Quadratic Field with defining polynomial $.1^2 - 2 over the Rational Field
[
    <$.1^4 + 1, 1>
; o) v6 A7 I! v1 _]

8 _3 i! o5 G0 J% p, t, d' I. jR.<x> = QQ[]
F6 = factor(x^6 - 1)
5 O+ g2 F2 O" m) Q8 QF6

- W1 ~- g' V9 C$ s4 Z(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;
. Y/ s. }. \" _1 `/ CC:=CyclotomicField(6);C;
FF:=CyclotomicPolynomial(6);FF;
. B" M( @; ?* B( o0 p, t
' T2 p8 }4 O# k5 qF := QuadraticField(6);
. I' l% x- ^) P' l9 O" \F;
D:=Factorization(FF) ;D;
1 o; Z( c* R4 k- m* c& ~Quadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
Cyclotomic Field of order 6 and degree 2
$.1^2 - $.1 + 1
* f$ ]7 ]" x& A0 h$ s- X! v. o7 KQuadratic Field with defining polynomial $.1^2 - 6 over the Rational Field
[
    <$.1^2 - $.1 + 1, 1>
]

R.<x> = QQ[]
  G8 W9 |, D, q. j! }F5 = factor(x^10 - 1)
F5
0 Y+ g5 |! |% K4 c5 G6 _0 c(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x +
  n+ `; T# f' _4 q& K3 w1)(x - 1) * (x + 1) * (x^4 - x^3 + x^2 - x + 1) * (x^4 + x^3 + x^2 + x + 1)

Q<x> := QuadraticField(10);Q;
) ^2 T: I/ f9 K1 g& w0 E& pC:=CyclotomicField(10);C;
FF:=CyclotomicPolynomial(10);FF;

; n" \9 |% p$ _( V, PF := QuadraticField(10);
8 [& T8 K- @' K/ l1 MF;
+ ~  [5 Z) X6 g* Z4 jD:=Factorization(FF) ;D;
/ ]6 _, U3 h0 l" L3 q  k2 `6 o5 eQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
+ L- U1 o6 j1 c3 t2 P7 z  P% H5 y2 @6 fCyclotomic Field of order 10 and degree 4
$.1^4 - $.1^3 + $.1^2 - $.1 + 1
; g6 p, W* v7 F$ kQuadratic Field with defining polynomial $.1^2 - 10 over the Rational Field
+ @. D% _3 X7 a: i! W3 m  Y6 n[
    <$.1^4 - $.1^3 + $.1^2 - $.1 + 1, 1>
]

lilianjie

7 ]- Y6 T5 [6 W5 A5 h7 q

分圆域：
C:=CyclotomicField(5);C;
3 E6 c$ r% ^1 G5 r: o) L! O" CCyclotomicPolynomial(5);
. A; v' ?& a1 G( v7 Q4 WC:=CyclotomicField(6);C;
5 S$ S) y) R. v1 CCyclotomicPolynomial(6);
CC:=CyclotomicField(7);CC;
2 @+ [( ~& B7 h# f8 HCyclotomicPolynomial(7);
2 ~, r3 T, ]  F( W9 ?MinimalField(CC!7) ;
MinimalField(CC!8) ;
' ^- K9 A; y" K# l" t! _! D* y: FMinimalField(CC!9) ;
5 C& p7 k( A# P. JMinimalCyclotomicField(CC!7) ;
RootOfUnity(11);RootOfUnity(111);
Minimise(CC!123);
; x7 _. K/ `% uConductor(CC) ;
CyclotomicOrder(CC) ;

CyclotomicAutomorphismGroup(CC) ;
6 V. o2 x" k  c) |9 j5 k
  Y5 z0 @! ?/ W7 TCyclotomic Field of order 5 and degree 4
& K! E2 a  `( }* ^2 c/ `7 ?$.1^4 + $.1^3 + $.1^2 + $.1 + 1
. e9 N1 J- }! M! b7 C5 gCyclotomic Field of order 6 and degree 2
$ ?; r9 p$ j8 e8 ]$.1^2 - $.1 + 1
Cyclotomic Field of order 7 and degree 6
& n8 j5 v- R' d+ ~) _. W" o; e$.1^6 + $.1^5 + $.1^4 + $.1^3 + $.1^2 + $.1 + 1
Rational Field
Rational Field
* A4 `1 O% D) }Rational Field
9 O. W- S' G2 Y) ^( |3 wRational Field
5 Y, G+ y4 a* _8 azeta_11
* E. h: I7 l$ D& J  Gzeta_111
123
- w; C% A/ }, J2 c7
% L& H3 u" C$ o& A2 z( c3 p: l+ @7
8 w. h) v( m# q+ n' Q$ p1 aPermutation group acting on a set of cardinality 6
) s! B% `# s( D! @& @  E, z: F: X8 wOrder = 6 = 2 * 3
3 k3 `* p) u, y- `    (1, 2)(3, 5)(4, 6)
5 m" |4 J3 A3 v    (1, 3, 6, 2, 5, 4)
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
CC
5 k4 Y* _* {0 `. yComposition of Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to GrpPerm: $,
) F/ ?  s' Z; O0 \9 CDegree 6, Order 2 * 3 and
Mapping from: GrpPerm: $, Degree 6, Order 2 * 3 to Set of all automorphisms of
1 `4 i' V6 v/ ^7 ^CC

lilianjie

F := QuadraticField(NextPrime(5));

KK := QuadraticField(7);KK;
K:=MaximalOrder(KK);
, j* G& T# G9 p& f& p) U( i# Q1 t3 wConductor(KK);
ClassGroup(KK) ;
8 y) F( a3 ~/ T. g( g8 sQuadraticClassGroupTwoPart(KK) ;
8 |  ~5 Q! b2 c" u2 m7 F$ cNormEquation(F, 7);
A:=K!7;A;
B:=K!14;B;
& _3 \* h2 M* s$ t: L. a$ R# p% `Discriminant(KK)
( I1 Z+ P. O. T$ u
' O+ e: R5 f5 S* S2 A+ b% `Quadratic Field with defining polynomial $.1^2 - 7 over the Rational Field
28
Abelian Group of order 1
1 e! ]0 {, R/ [' A) lMapping from: Abelian Group of order 1 to Set of ideals of K
8 m  j$ [% O, U+ ~. aAbelian Group isomorphic to Z/2
  F# f0 p% B6 _3 T& g0 `Defined on 1 generator
Relations:
2 ^% g2 {$ p9 j& z, |: x    2*$.1 = 0
2 W# p* P7 I* g& Q( `# L/ [! BMapping from: Abelian Group isomorphic to Z/2
Defined on 1 generator
Relations:
% \: O% L, V1 f* O) O0 o    2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no
inverse]
false
7
14
( U+ Y4 Q; I9 C8 A8 G) @28

lilianjie1

pell   equation

lilianjie1

Dirichlet character

lilianjie

" }7 G5 E: M& U- G" x' a3 d
8 n) v/ M3 m1 h! ODirichlet character
7 J, A' J# g5 hDirichlet class number formula
9 s& Q# |& P, w- v6 f
2 A2 D; e8 M' A# @% f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

-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

7 A$ W; K! _( z" q-3时  6个单位根                             N=3  互素（1，2），    (Z/3Z)*------->C*         χ（1mod3）=1，χ（2mod3）=-1，
- Z9 u6 _: R* Y* |2 Gh=-6/（2*3）*Σ[1*1+（2*（-1）]=1

3 h0 @. k5 ?  M& w  o' n-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，
/ [& O! y2 u. p4 x: [
, U& C& ^) u/ e( v2 G7 o

9 a3 w  `  \8 ?1 l* N. rh=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2
& Y8 b2 w8 F- Q! n, B" }5 w

* t# o. ^( c# t) e
-50时  个单位根                          N=200
0 e# ^/ t; e3 H( j+ O' t8 V+ D( J

孤寂冷逍遥

lilianjie

看看-1.-3的两种：
: q1 n0 e6 z! b1 J% Z5 k' s; V5 E
- n8 R9 x% T( {6 {: x; |Q5:=QuadraticField(-1) ;
Q5;
7 v2 k- c+ W( Q; N% b; x1 C
Q<w> :=PolynomialRing(Q5);Q;
/ ?, s/ U' V* y0 w" b: n! GEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
M;
% m5 u+ u7 d/ d7 n. pNumberField(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  x4 P6 ]% C, ~2 M2 H% cIsQuadratic(Q5);
IsQuadratic(S1);
+ Q- v$ s; X2 ]/ ~IsQuadratic(S4);
, P, x# ~! o, m5 b6 T/ d1 k6 CIsQuadratic(S25);
IsQuadratic(S625888888);
* p: I( M/ Q1 I9 p/ @3 F2 [) C. BFactorization(w^2+1);  
7 ?+ f- B( ^( q7 bDiscriminant(Q5) ;
FundamentalUnit(Q5) ;
FundamentalUnit(M);
Conductor(Q5) ;
( k* ^. U) T9 b% T4 w& G6 h
( w, i: m0 [( `Name(M, -1);
8 d; p: t- h- O" z- lConductor(M);
ClassGroup(Q5) ;
& v2 A4 X  [- ^ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
2 T/ O; g2 D( j  H3 J7 HPicardGroup(M) ;
( n8 W& W/ g& M* k! m; v( TPicardNumber(M) ;

QuadraticClassGroupTwoPart(Q5);
QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -1) ;
NormEquation(M, -1) ;

Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
* n3 x, s' _+ Z, ~Univariate Polynomial Ring in w over Q5
Equation Order of conductor 1 in Q5
Maximal Equation Order of Q5
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
Order of conductor 625888888 in Q5
2 C. d: o9 j: x  |# n( wtrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field
true Maximal Equation Order of Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
true Order of conductor 1 in Q5
3 H8 W- Q1 B' e( a+ s! U+ ?[
. K/ g$ d$ @* i3 \& B) ]    <w - Q5.1, 1>,
/ c" X( C6 y: j/ E: H% A. p  T$ g    <w + Q5.1, 1>
  G# N; g6 d0 j7 G]
-4

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
8 I+ i/ J, b8 ?0 T3 Y- r( P) \  K& B

5 [* q4 D+ W7 x7 f>> FundamentalUnit(M);
                  ^
1 E9 R- t' F, T6 [Runtime error in 'FundamentalUnit': Field must have positive discriminant

4
6 a6 V2 F" F- A3 I! }' J- a$ h5 N
>> Name(M, -1);
       ^
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]
/ f- R2 z- ~3 l6 r& A
1
2 W5 a: M3 g$ t3 c) M4 \Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
& ?  K! R. R0 U8 ]* \/ A  _Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M
5 i8 X" c3 q( b8 ]1 H1
1
) w" P: R4 ^$ W5 z$ vAbelian Group of order 1
" _- O* O: z5 y4 t. c- d6 \Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
( w3 Y# Z! r$ {0 [3 G1
, R# V- [$ z# Y' RAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
-4 given by a rule [no inverse]
6 u4 n/ L1 ~3 L* s" W9 MAbelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
" K, v. F% J8 v# S: L-4 given by a rule [no inverse]
false
false
===============

Q5:=QuadraticField(-3) ;
! \0 u8 u# r. `) t7 fQ5;

Q<w> :=PolynomialRing(Q5);Q;
0 c) ?4 s) b/ |" hEquationOrder(Q5);
M:=MaximalOrder(Q5) ;
! q2 u/ {  \& X# |; J7 gM;
' B; k5 F; R5 |, ^/ KNumberField(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);
* K1 |+ e5 y( a/ v- _- g! I/ hIsQuadratic(S1);
4 l" `5 w( u! p2 @& Q9 O3 q; v. YIsQuadratic(S4);
9 h2 _! l; N  Y6 a$ A' `! V6 }IsQuadratic(S25);
; F& H. f( L6 g0 N3 ]( _& Y) ~6 \IsQuadratic(S625888888);
) |7 S: c% r. v; s# oFactorization(w^2+3);  
Discriminant(Q5) ;
! m% c% m. B! e2 bFundamentalUnit(Q5) ;
9 r4 ^! |6 \/ u& i$ ^FundamentalUnit(M);
8 ^! e4 |! f% J, v5 \- mConductor(Q5) ;
5 o  I8 o( o- ?% X6 K1 d
Name(M, -3);
Conductor(M);
. o' o! E* N; y% g) bClassGroup(Q5) ;
ClassGroup(M);
ClassNumber(Q5) ;
ClassNumber(M) ;
! T6 U4 i$ u. |1 Z8 F$ B5 \PicardGroup(M) ;
+ F: Y; f# I5 _" l& Y" V* k( DPicardNumber(M) ;
) e: H3 h! q* o  _
( s& E% J7 _2 e/ E% j9 M) M' |9 m$ DQuadraticClassGroupTwoPart(Q5);
9 @4 U4 j" [0 s6 M$ P# K* [QuadraticClassGroupTwoPart(M);
NormEquation(Q5, -3) ;
* L8 k: T# Y0 ENormEquation(M, -3) ;

Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
5 O2 A( h% u3 d9 `& dUnivariate Polynomial Ring in w over Q5
Equation Order of conductor 2 in Q5
0 i5 ?" v7 C  F) p( OMaximal Order of Q5
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
Order of conductor 625888888 in Q5
) {: |4 j2 b) k; h' V; \% [& Vtrue Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field
true Maximal Order of Q5
true Order of conductor 16 in Q5
true Order of conductor 625 in Q5
5 s8 [7 M) m! O% ~# jtrue Order of conductor 391736900121876544 in Q5
: t4 L# F2 k: _8 k% Y- d[
  x2 N! K  ^5 m2 s* W+ f- C; P    <w - Q5.1, 1>,
    <w + Q5.1, 1>
+ A5 c$ F" l) y" O0 K: `: A]
-3

>> FundamentalUnit(Q5) ;
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant

: w. u- K4 H. p" j' U2 y
9 t0 P9 Q. K4 A7 O. c( ^0 W" _% N4 J>> FundamentalUnit(M);
                  ^
Runtime error in 'FundamentalUnit': Field must have positive discriminant
* ^. O! y  u1 T. L' S: G& c' k
3
! {3 G; B# b7 o% R$ O2 H" k2 Z+ k5 h
/ A; k$ J. N! ~6 M6 I>> Name(M, -3);
       ^
7 e$ S1 ~$ m) c7 g" P" O2 ?Runtime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]
- g2 [5 u5 y9 C7 G6 D- N
1
Abelian Group of order 1
* m( H/ G" O2 ]- t+ J% x6 DMapping 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
1
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no
inverse]
1
+ }( k; y0 L8 d4 AAbelian Group of order 1
5 |# A* u# V6 D! fMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
% s) N: H( T& R4 Y0 W) {9 y" D-3 given by a rule [no inverse]
Abelian Group of order 1
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant
6 r8 t" M" I: {7 k-3 given by a rule [no inverse]
false
false