' X6 q8 t+ U {" e$ B: ]Q<w> :=PolynomialRing(Q5);Q;. e" p, Q6 _/ ^" e8 I; Z1 ?9 Q
EquationOrder(Q5); x( T+ a7 d+ Q' A9 d7 D" Q2 }M:=MaximalOrder(Q5) ; : [" u' M' i! n1 c% dM; : A" h5 l3 _; L; r* xNumberField(M);6 }8 P& l2 F3 H) ~7 a
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;" g5 S* g, G0 M% ^; ?. L/ }0 ~
IsQuadratic(Q5); " J: z& O2 |5 F! h% R. QIsQuadratic(S1); & j" }/ x8 I& S# I8 ?; T1 x; }# LIsQuadratic(S4);' g/ I( L* R1 T6 G, g
IsQuadratic(S25);3 Q) ^, f( J. z, V( [$ c
IsQuadratic(S625888888);/ ^! a1 E7 P% a; P8 P p0 `8 f- Q
Factorization(w^2+5); ) X( i7 B9 x2 R: iDiscriminant(Q5) ; 0 ~1 g, V0 p3 ]# m. _FundamentalUnit(Q5) ; ! z% F$ p0 u, [FundamentalUnit(M);0 l7 ^7 A# Z; t: G+ r5 C
Conductor(Q5) ; % k# H5 B! M- m! V8 R; e 2 e& x& s. h- WName(M, -5); ) K0 h3 { G2 g, x' MConductor(M);6 B4 P* e$ S' C% f
ClassGroup(Q5) ; @, w9 l; @( W6 H: `1 }# m
ClassGroup(M); 5 B. a/ s! {3 U1 o: jClassNumber(Q5) ; & e1 G7 d1 o! K) R [6 t4 yClassNumber(M) ; ' B! H! h/ t5 @, y3 {" WPicardGroup(M) ;, K; t: g2 M& U1 C
PicardNumber(M) ; 8 k$ k) r! } ^+ ~( s/ q) [3 s5 y; w
QuadraticClassGroupTwoPart(Q5);& B, X3 Q0 Q, q* U6 e
QuadraticClassGroupTwoPart(M); / [+ N5 M) I; ?4 E/ rNormEquation(Q5, -5) ;# ~# A: a0 {, b# F) J
NormEquation(M, -5) ; . Q. p* Z! s6 Q [9 bQuadratic Field with defining polynomial $.1^2 + 5 over the Rational Field 8 z. n* y( p& P" D+ bUnivariate Polynomial Ring in w over Q5- M. g# ~+ v8 m; k/ i
Equation Order of conductor 1 in Q5 ' e! C- @% Q9 I$ J' e( c9 rMaximal Equation Order of Q5" e8 e3 q% }8 K, i* e/ M
Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field" |0 @+ F9 Z t! @; Y# ^- w
Order of conductor 625888888 in Q5 6 V6 r) Q; \6 M6 htrue Quadratic Field with defining polynomial $.1^2 + 5 over the Rational Field( @6 v Q3 f) p$ T! a( m0 t
true Maximal Equation Order of Q5& D2 Q$ ?: q( V/ C) f# f
true Order of conductor 1 in Q5 ; M1 w1 v1 r- Q; ~true Order of conductor 1 in Q54 ^* a3 D6 Q9 G. [
true Order of conductor 1 in Q57 J: l7 q+ n0 F+ {2 Z0 G X
[ + u* R# S H% z' { <w - Q5.1, 1>,# k$ C9 Z% F O) I3 j
<w + Q5.1, 1> ; H7 p; Q; _5 F+ D0 A4 q' S] U& C- Z! q ^0 W
-20" N0 c$ {8 d' p# H( m* ^7 ^7 u# d
; |/ i% o! v5 u1 L+ v
>> FundamentalUnit(Q5) ;6 T7 l, U" v7 L$ T
^, \3 @. F% o! H
Runtime error in 'FundamentalUnit': Field must have positive discriminant ' m) D# h* H+ J) F0 o: e7 u. T4 A, R$ I0 j. J: {. T2 |2 N
: y7 y4 R$ b* t* ~. P% J* h0 ~7 }- |. u
>> FundamentalUnit(M);% S7 N, E4 z% Z" c- \
^ ) ]- K0 _! h9 n' t" W cRuntime error in 'FundamentalUnit': Field must have positive discriminant 7 s5 l8 F7 ^0 g7 K7 `. I, I C8 `6 ~& K4 K9 {2 }
20! W' h+ x) \, X6 X0 R; _2 T% X
2 J5 V3 i8 j# s
>> Name(M, -5);3 I% u8 |8 x- x% ^
^ + a; Q* C, N# E: H+ y' A, e; pRuntime error in 'Name': Argument 2 (-5) should be in the range [1 .. 1] ( K) o+ Y# ?0 S ( f: C) O! x3 `% U# g1 : [, {9 P0 u0 M8 Q. p4 ?- A9 m% \Abelian Group isomorphic to Z/2 ) q) @: U) o* M8 O5 @& e+ ZDefined on 1 generator- X4 H+ c: s9 W4 H4 r
Relations: 9 G8 y7 f. N0 W/ Y, R 2*$.1 = 0( L4 U' s& `' h' K, d
Mapping from: Abelian Group isomorphic to Z/2; {1 J7 x# u0 K# a! m+ W
Defined on 1 generator $ ? R$ t+ q- P1 X) I0 aRelations: 8 C1 N4 N/ l j* ` 2*$.1 = 0 to Set of ideals of M+ I1 Z+ H9 S& A2 `' i# D
Abelian Group isomorphic to Z/2 # _6 l4 ~+ A4 C' Q6 nDefined on 1 generator2 z; `, t" S/ H
Relations:% f/ g6 q+ W* t, I. i: I
2*$.1 = 0: s/ \% _! q: X) [% y
Mapping from: Abelian Group isomorphic to Z/21 P0 Y$ Z$ Z9 c( I* c' B
Defined on 1 generator+ |! R3 L3 C; G* q
Relations: & R5 i* C* ?) O+ j/ L7 k 2*$.1 = 0 to Set of ideals of M 2 H+ D2 Y8 f0 j* i2 ]+ r7 [2 # y" N( Y5 K' i0 r2 ?; }& }9 _$ ^0 [2$ t9 [" Q# x2 O* V0 Y4 P
Abelian Group isomorphic to Z/2% `7 k6 _( \* n" ]/ m& z* L Y
Defined on 1 generator! n- n# T7 W* |% `7 {9 P
Relations: / S! Y1 j5 B) P7 \2 p* Z X 2*$.1 = 0 0 S4 V0 t* g; v5 k- n2 gMapping from: Abelian Group isomorphic to Z/2 , w, D% I* R0 F( ~" i, hDefined on 1 generator$ e8 Z* X! M2 c9 I- }; z/ o! y
Relations: 2 ^, l) j# D" y 2*$.1 = 0 to Set of ideals of M given by a rule [no inverse] , Q3 `3 w! {+ u2 * P4 M4 i! q9 L+ p$ BAbelian Group isomorphic to Z/2' a6 m4 b/ |# ]
Defined on 1 generator " R9 x6 x3 I6 P. @8 c3 G$ c1 VRelations:7 H9 d4 O5 `" F8 X0 M
2*$.1 = 0/ V7 P$ W2 y+ }+ W" V7 b
Mapping from: Abelian Group isomorphic to Z/2+ G5 ~& K# i. ^5 w6 x: h0 C* a* G& x, E
Defined on 1 generator ' z F) M4 m/ c3 bRelations:( ^. D9 [5 D1 i/ n9 e, h4 Y$ c) a7 p
2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no . s! l4 I. j" d8 c4 f1 Z( w9 Iinverse] 3 Q/ U7 @" F' p) g$ EAbelian Group isomorphic to Z/2 6 H9 [4 M- Y/ Y, RDefined on 1 generator & W% n0 v$ Z' K& }Relations:, \& B2 R4 Y5 F7 ~3 h1 g& G6 O5 K
2*$.1 = 0 `9 u# h! k. a0 z9 T
Mapping from: Abelian Group isomorphic to Z/2 : }$ k. e8 a$ `8 eDefined on 1 generator- o& v# j1 B( S/ r% `3 E& R
Relations:, v* h* y# l9 [6 `5 b
2*$.1 = 0 to Binary quadratic forms of discriminant -20 given by a rule [no - N* t6 n6 }& p/ xinverse] d z- X) a4 u rfalse 0 z p7 o" @" l7 [* Z; W9 f# {false/ @- A& K) K2 E! I+ o; A0 J: n k
==============+ ]" ?; k8 f; N4 f$ d; @" q2 G
& X+ D1 T; W+ W
" n5 M( m/ j4 m! `' f& l
Q5:=QuadraticField(-50) ;; V \# ?: H1 N H
Q5;7 \0 p+ M1 j @3 U
: y4 ], a. P# M* o0 F( i
Q<w> :=PolynomialRing(Q5);Q; ) ~8 s' C+ D& J6 eEquationOrder(Q5); . c+ b2 y$ e: s5 Q. x6 f aM:=MaximalOrder(Q5) ;. [6 b7 r* o- ~0 E
M;( s F- U9 w6 [) Y9 l; h
NumberField(M); 2 n9 X Z7 l/ N, c0 lS1:=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; " I7 ~( `/ E8 z1 Z3 ]1 s$ eIsQuadratic(Q5); " ^. Z. q0 I$ ?6 m' dIsQuadratic(S1);! S& ?6 f- g7 U; c$ f
IsQuadratic(S4); % y" H& Q- s9 ]' c0 AIsQuadratic(S25);, S- X% x. I- t0 s% h. P
IsQuadratic(S625888888);$ a% A; A- Q) t
Factorization(w^2+50); . ?$ o$ v2 @9 m: ^# i3 B" M( m/ z' u9 r
Discriminant(Q5) ;' R2 q# @/ P- s/ L5 _
FundamentalUnit(Q5) ;: j5 _7 ^$ ]! |. |9 [6 z! f3 }! m
FundamentalUnit(M);- T9 V1 v2 n- n! H& D; L) E
Conductor(Q5) ;& X6 v3 U- C; Q7 P! L0 Q. }6 k w$ J
_* M, M/ l/ s1 }% O( gName(M, -50); g' W# L( ?* _$ j( u
Conductor(M); 2 y) l' {9 O. g5 \( }9 ~ClassGroup(Q5) ; : H. F2 z& p6 D) z2 d
ClassGroup(M);# w& b7 A! o* z/ L/ Z; Z; Y9 l) Q F
ClassNumber(Q5) ;) z# E% d0 ~9 u" O, @8 T4 B. O
ClassNumber(M) ;4 f% j( W/ C) g8 {
PicardGroup(M) ;2 i. i- s; {' X+ }
PicardNumber(M) ; 1 ]. Z1 ], ^9 d3 M2 N4 X0 `3 [$ c! l- V$ Z# C1 e8 t
QuadraticClassGroupTwoPart(Q5);# p: n+ G' k# H% W9 J' m `6 W/ U+ F8 }
QuadraticClassGroupTwoPart(M);8 U) j% Q. Y, Q5 v% M
NormEquation(Q5, -50) ; 6 p+ h; b+ W+ VNormEquation(M, -50) ; 0 A1 G* e) }# d2 e* D : y0 s# C! ^5 C4 |; wQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field ) }! A) E2 ]. y' h# }& U$ sUnivariate Polynomial Ring in w over Q5 " s9 R+ i* B7 K3 |6 ]$ d5 oEquation Order of conductor 1 in Q5 4 l3 T6 i; V# mMaximal Equation Order of Q5$ U1 u; u1 p' D& a1 w
Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field 6 Y+ F: T4 W" Y4 {! SOrder of conductor 625888888 in Q5/ b3 T6 @6 H k* {) e0 C, @
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field ' W' W2 J0 A: W/ {/ Qtrue Maximal Equation Order of Q5 / T: Z8 n- ~, }& v- U% l: |% O1 gtrue Order of conductor 1 in Q5' @8 s- V q8 d6 f7 o
true Order of conductor 1 in Q5 4 f' m' d) I' j4 n& Ytrue Order of conductor 1 in Q5 3 {1 J% E+ y- F$ k[6 B1 d; B# G" J( U
<w - 5*Q5.1, 1>,2 y ~! ~6 n. p8 U, ^# {; S9 i
<w + 5*Q5.1, 1> L& [4 p% |9 `! e" W B/ t' B
]+ y* t6 D0 O9 a i: h' R2 j# ~
-8# Y# r) |5 n5 `
/ R6 k% e9 c! y5 p+ V>> FundamentalUnit(Q5) ;7 _- L: V: ]* y% e" r
^+ i8 s! y \ S
Runtime error in 'FundamentalUnit': Field must have positive discriminant. }" p- n- n; }0 P3 W
0 y7 k E1 j5 s& j# G+ W4 y& c # b/ ]# j2 m) k* @>> FundamentalUnit(M); 0 H; C+ `2 ~5 F' e0 q2 ^ ^9 n8 b: T' {. p) R& c- p# ]: ]
Runtime error in 'FundamentalUnit': Field must have positive discriminant; h7 Y( N8 P+ t2 c$ H, g# J7 q
$ q% z, t& @* ?5 n7 R6 H- ^
8 5 ~! j7 o" F0 {! n( W! ]7 B; j6 O# D* U! s
>> Name(M, -50);# G1 c! u' K, l Q
^ $ Z# Y9 D5 W' j6 M- c, `Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]( R* C' h) {) S) c, B
. C" j) ]5 K: T# {! a1 ; b; U' f/ p7 G6 N& n" a; g4 DAbelian Group of order 1 , S; J0 X; G" k0 C% k3 k% Q0 o* dMapping from: Abelian Group of order 1 to Set of ideals of M) T% o* v4 E/ L2 |
Abelian Group of order 1* \' N6 |) m# |% h- \
Mapping from: Abelian Group of order 1 to Set of ideals of M6 N0 e" E( ?0 t4 D8 m
1" x* u1 y: M' Y& _, \
18 g) l7 @- _0 W1 B1 c, ^2 m
Abelian Group of order 1. L' b5 t2 |* f
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no " l" n3 Y6 _5 H' tinverse] ; Q7 [4 N( M, @1: [5 @* j4 I: O' D7 B
Abelian Group of order 1 ! v a4 T) Z1 sMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant , ^) T) f- l7 Y2 c-8 given by a rule [no inverse]- o: w4 D5 _, E" H
Abelian Group of order 18 a2 Z" [# H4 Q9 n+ W$ X
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant * h6 ^* w/ n" u4 ?-8 given by a rule [no inverse]3 ?, k& ?) w( u" S4 ^) R+ z. d+ h
false ) S" i( Q7 H/ t6 t, Hfalse- U% T( j! o9 r; g( x3 \; L8 j, x- h
! U5 Y3 H! Y$ @! h- SQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field 0 b' c% b) i/ ?3 @# p* r6 X; KUnivariate Polynomial Ring in w over Q5; u4 X/ B% j3 n' U2 M s0 {2 I
Equation Order of conductor 1 in Q5 4 K$ ?8 g/ [7 t4 d/ m! e% \Maximal Equation Order of Q54 b% d( \6 j3 \0 B& d
Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field ) s. b% M$ c7 G4 y5 ~Order of conductor 625888888 in Q5 * v& N5 I z6 b9 R5 V& \( p9 rtrue Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field ! ~& n1 H) E8 h" h& `/ r4 vtrue Maximal Equation Order of Q5 a6 I6 x! V1 E& \9 G: b
true Order of conductor 1 in Q5 # s9 H+ \/ g, ?0 k3 F' e5 ^2 i6 D/ ftrue Order of conductor 1 in Q5* {! |, J2 U( g* n
true Order of conductor 1 in Q5 ; }" x% w" a& ~9 b. o- o) P0 E[ 7 i {' h" O. ^8 r; J: U <w - Q5.1, 1>, 1 L! d' y0 U* }* J3 L <w + Q5.1, 1> & X7 }# V, D, ]1 C" G2 V] 4 v, m ?7 j _+ s& M- ?0 J-41 m8 u. }1 M; ]+ e$ b Q
# ?$ Y: r' f! U5 {) d9 i>> FundamentalUnit(Q5) ; ' V* |" j# Q8 U4 ]8 f2 O7 L ^8 r5 F. O0 C8 D% ^( @6 s/ r( R
Runtime error in 'FundamentalUnit': Field must have positive discriminant 5 o+ p' i7 p( B5 h& I1 ~ $ q1 C! c( ]8 p) M* n% {- Y% W# h7 G7 c# ]/ _# I2 E$ H
>> FundamentalUnit(M);1 }8 K& u1 T0 u* q1 q; ?- h' h* X
^ # k/ D& ^; ^" U. d5 W' ~Runtime error in 'FundamentalUnit': Field must have positive discriminant ( Z9 p( \4 k6 N# }5 M Z, \ + X; E! t& K. K+ V2 W# d' h4% ~6 |2 D! N: }' |' Z2 f: R
6 p K$ G5 }3 |. b
>> Name(M, -1); 7 Y3 D& s; |6 X ^ 4 M& T0 | g% j* I1 d& ?1 MRuntime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1] & }% Z" N. G, j& }9 g) L1 A! O' X5 r: T& [* g( g3 m9 J
12 N, W/ E2 D& f! K) w1 ?4 y9 P
Abelian Group of order 1 % x% X9 e- j( ^1 |! qMapping from: Abelian Group of order 1 to Set of ideals of M3 y6 }7 k0 A9 U4 {+ S) C
Abelian Group of order 1/ d5 D: a R4 n
Mapping from: Abelian Group of order 1 to Set of ideals of M ) b3 T9 f5 G' s) v1 z2 s [1 v1 f! K9 Y# S7 B5 g+ i. I% C
1) O* f5 s$ \0 A2 ]2 ^
Abelian Group of order 1" k" l" @# w# e6 i
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no9 e6 p( K7 P$ j( u" n6 F
inverse] ! L- q" v1 _2 E1 z17 e# _8 g7 h) M2 R( }2 {( |
Abelian Group of order 1 9 D i6 |/ x( r: _1 M" t" I% UMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant+ N6 I! \7 p' t( B
-4 given by a rule [no inverse] & w8 T6 F2 \+ ?Abelian Group of order 15 a6 [0 U- X) @
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant! P. n2 K3 R$ l* C
-4 given by a rule [no inverse] 2 Z! h* M0 n$ C0 P" D$ I% afalse/ @) d& b" w# q. i4 r5 Q7 e
false* ~# a; I. q9 o4 m1 K
=============== $ h; Z$ E4 p) x4 j" C# R" ^/ X ) O5 d! u5 X" l9 `7 k F, _$ qQ5:=QuadraticField(-3) ;+ G8 N7 U& I+ z5 P8 j1 U$ u
Q5;, U% L. P+ c: j m8 z2 K
/ e* Q) f7 ?8 B/ v @/ uQ<w> :=PolynomialRing(Q5);Q; s6 [; E; d" s$ t3 d# \EquationOrder(Q5); & H8 F/ g+ f! l, RM:=MaximalOrder(Q5) ; 9 r3 v3 f& q% @5 |M;6 U4 h I5 T0 @7 a0 Y- w
NumberField(M); % y8 K9 L) n2 ~4 j8 l& US1:=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/ `8 T5 }: }. `" J! e" QIsQuadratic(Q5);# I; X0 J" f* h3 d7 I' S4 r& U
IsQuadratic(S1); 4 o+ l" ]0 x, `) ?& p9 o& m% t5 e" DIsQuadratic(S4); / j& e; |! W& @7 h i: iIsQuadratic(S25);3 k- [* T6 N% v* q, m
IsQuadratic(S625888888); # s/ v6 q- v& q5 Q: tFactorization(w^2+3); - |( w8 }0 ]6 U# c4 }Discriminant(Q5) ;! r# J% D& v9 h' K7 t9 ^
FundamentalUnit(Q5) ; 4 F4 L* ?/ V1 M6 }* }) ^: s" RFundamentalUnit(M);+ y/ C5 N- H9 I/ D! h8 y, \* i$ I
Conductor(Q5) ;; H* K s) ^6 j; j' R
# `3 t! D; n2 Q
Name(M, -3); 9 S+ P4 q- H9 t- }Conductor(M);( j- i' q4 r4 X5 e1 e8 a: T1 C3 [
ClassGroup(Q5) ; n9 ~1 D" G2 ~ClassGroup(M); : h7 m4 u9 X1 W; lClassNumber(Q5) ; 1 ~& d7 B/ N/ m9 g; {& V% S% E% HClassNumber(M) ;1 a. h8 u' E, y& I4 p
PicardGroup(M) ; ! Q, B x' F$ JPicardNumber(M) ; ' n5 M c+ A+ x, }9 j+ A# H7 F, B; f; D* i: o6 V. W/ K
QuadraticClassGroupTwoPart(Q5);' \5 a! l$ o1 g! l0 k
QuadraticClassGroupTwoPart(M); / O9 d7 c' K5 F) xNormEquation(Q5, -3) ;% {* E; S% g" z! p$ l
NormEquation(M, -3) ; ' K$ B4 F4 c, w- ~8 i : P V5 D" | H) ?2 i; C; [Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field + |+ O" u9 @6 d4 kUnivariate Polynomial Ring in w over Q5 % Y, B) u6 i* PEquation Order of conductor 2 in Q5 . `! Q" N+ Y7 v' BMaximal Order of Q5 ) ~# @' N. l) b6 CQuadratic Field with defining polynomial $.1^2 + 3 over the Rational Field6 s( K; V8 p0 }
Order of conductor 625888888 in Q5 : j2 ]. b8 z- T; R- M3 u1 h6 v0 P7 Y) _ ]true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field9 r( G+ U: G9 h4 L8 q3 [$ m
true Maximal Order of Q5 ; l% L1 k, y# Ctrue Order of conductor 16 in Q5 . ^$ ^6 p, v' Atrue Order of conductor 625 in Q5 ; n7 Z7 G2 J1 ^true Order of conductor 391736900121876544 in Q5 : p8 P2 P- z# S; w7 h* v7 \3 }[( O6 {/ p$ L, f5 J8 V# ?3 a
<w - Q5.1, 1>,0 u8 V' j6 x% W# _( m! F
<w + Q5.1, 1> * N/ k* Y* x2 G4 s6 C]$ L; l! z) M9 N- W7 N0 m) `! |
-3+ m1 h: ^( k$ @$ g
) n( E" ?* K- n( d3 z3 u
>> FundamentalUnit(Q5) ; 3 h) s& x, k) p ^3 w$ b& T% P- N
Runtime error in 'FundamentalUnit': Field must have positive discriminant + |) e! G; k# Q+ A) C- d, S# _& B' ]
. N% s! u9 y6 E( r! R' u' j>> FundamentalUnit(M);5 C9 l1 T# h- _4 a: {5 ^
^$ d/ v$ k9 m; U: {! p
Runtime error in 'FundamentalUnit': Field must have positive discriminant" h G U) Q/ P/ I/ {. I# [, K
6 |% u* S8 @" L: `
3$ d4 ~7 {. s' {8 C2 a! }9 d
- |+ `0 p- l8 K. A# z$ @3 e>> Name(M, -3); 5 v+ N7 ^4 N$ f% p8 \ ^ & _4 k$ M4 }9 H' s. HRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1] 8 V6 T4 I7 U u7 b' s * _8 q# q, h" ^- `; T' E1 2 f( k) b, C( K+ G$ J+ HAbelian Group of order 1 & c# K/ u, ]6 d; n& }Mapping from: Abelian Group of order 1 to Set of ideals of M$ A, g& E. c$ m. H$ d9 F
Abelian Group of order 1 : I. P5 t p eMapping from: Abelian Group of order 1 to Set of ideals of M0 C: e1 o* X! V/ W, V
1; L5 r8 B" Y/ V7 q" X$ n0 X h
1; m: E* T B! v6 {; H% ]# Q6 I1 Y
Abelian Group of order 1 1 O8 c) r( ? U `7 w5 V/ K& AMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no) j! T: f- f O" [' o/ N, j" m4 |
inverse]4 { q* }! ]8 ?- j" I$ A
1 ( {) H3 J, }- Z+ `) l7 kAbelian Group of order 1 ! `% o) g# ?5 J, e9 E5 t. ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant) E( E0 Z4 C2 x. J, N) I/ a' L) C
-3 given by a rule [no inverse] ' o: N2 Z$ N1 FAbelian Group of order 1& o; F# P1 V C8 ]
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant. O7 H. a& m4 G, _& }2 [+ P
-3 given by a rule [no inverse] : P; S* M, I, gfalse - y+ p1 R B' S+ K3 x+ Ufalse
本帖最后由 lilianjie 于 2012-1-5 15:20 编辑 6 c# s; E, |6 k$ a' a5 L
0 V: a7 i9 I# S8 m
Dirichlet character; z$ ^( u0 @; [8 b! S
Dirichlet class number formula& Z; x1 q% b, S Y/ ~1 P) x# c
- g; X6 X; T* g; z) @2 n0 {9 T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0 A y. s* g. Q; ?% A
! V1 I. Z7 @# S5 h-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$ Y [% s6 B8 X2 |8 E1 c8 Q
9 n, [( \6 U) x) ?" F) ~4 C) n/ q$ v-3时 6个单位根 N=3 互素（1，2）， (Z/3Z)*------->C* χ（1mod3）=1，χ（2mod3）=-1， U' \% \- q- q* |# C1 ^h=-6/（2*3）*Σ[1*1+（2*（-1）]=1 5 ]" k) ^( H- G! ^, W% Z6 n% V % e- X4 S" w( g. ?-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， / p7 L, a2 K, T' z0 S0 O& v % c0 _% _" @, [: g( L7 m0 M6 m9 Z3 h* c" T; |0 k
: `4 ~" I9 ?5 Q- u _$ x$ K
h=2/（2*20）*Σ[1*1-3*1-7*1+9*1-11*1-13*1+17*1+19*1]=2 2 r" y. ^" T d0 ]9 u5 S- z 0 U. Q, _- q5 r4 A7 K( k* f4 z6 f7 P- k: D
7 g1 i9 c# x' a6 l
-50时 个单位根 N=200 4 j: |8 h; E5 s3 f% s8 w
本帖最后由 lilianjie 于 2012-1-9 20:30 编辑 ) \6 T. F( ?! A9 c2 w
c5 {2 `8 n" ` ^% y7 XF := QuadraticField(NextPrime(5));2 c/ m m6 J* F& R
7 P- X) t& F ?
KK := QuadraticField(7);KK;4 f. B8 \5 s H2 j$ e$ U
K:=MaximalOrder(KK);) P+ p9 k% g8 S* h
Conductor(KK);( K/ }+ I# M! h; w
ClassGroup(KK) ;) n) @! i* m0 A
QuadraticClassGroupTwoPart(KK) ;4 Z% B" P" G6 N6 ?, N, ~# @, t
NormEquation(F, 7); ( ?" D% r/ e6 [A:=K!7;A; : T9 f' D5 a; L; NB:=K!14;B;) N# @ g8 ` f1 k
Discriminant(KK)& s" U+ a- X6 J+ [' u+ P) q `
, t5 O, @- D. i$ rQuadratic Field with defining polynomial $.1^2 - 7 over the Rational Field$ V1 h6 _0 U1 V- B, w
28. E- Y6 u; v. F3 t
Abelian Group of order 1 ( ^5 C/ s( e5 H E, i n/ p1 Y# }' FMapping from: Abelian Group of order 1 to Set of ideals of K $ S9 ~2 W7 `+ H0 A* KAbelian Group isomorphic to Z/2' Z. e$ A) X- \" N4 }$ \
Defined on 1 generator ) I1 J; ^7 _0 B9 {Relations: * M* N- w2 G; b" f9 m 2*$.1 = 05 Q9 \/ o" g# E1 ?9 x: t# ^
Mapping from: Abelian Group isomorphic to Z/2& W) f8 X" v" ^1 @8 f9 |
Defined on 1 generator9 Y8 U) T$ ], M$ N6 q
Relations:5 {. f7 y6 b/ O
2*$.1 = 0 to Binary quadratic forms of discriminant 28 given by a rule [no ( v! K' T; w7 m" S+ u0 hinverse]7 u; Z$ w/ N5 o4 D8 C5 f
false' r& b/ @$ B- v4 q6 O
7. }+ S5 J3 X! L; T
14 # u; B8 o7 x# N9 R. P! k28
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发表于 2012-1-4 17:41
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