' v3 ~4 C% j. @9 [/ g4 ]2 z1 l h( sQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field5 W! ~3 _' H- Z* D2 @
Univariate Polynomial Ring in w over Q5% |. _% C8 [" P( e0 M4 y5 t5 I
Equation Order of conductor 1 in Q5& e; j; {2 L. v2 e
Maximal Equation Order of Q5 5 t/ T0 H2 L/ fQuadratic Field with defining polynomial $.1^2 + 2 over the Rational Field . N6 }" v( Y9 b/ IOrder of conductor 625888888 in Q5) Z0 I0 e" r) Z( c* S" ^1 Y8 g
true Quadratic Field with defining polynomial $.1^2 + 2 over the Rational Field% X' `7 d( ?4 ], l9 Z: z
true Maximal Equation Order of Q54 T* c; p8 y% D* ^, n
true Order of conductor 1 in Q5 # H+ r v& p: L9 Rtrue Order of conductor 1 in Q5/ b' j4 f7 ~% J
true Order of conductor 1 in Q5$ ?' t; y/ @* l1 s3 v1 i. q" u/ b$ ]
[ & _+ o2 B2 x& B. c/ m! y <w - 5*Q5.1, 1>, , ]: L, M6 F2 x& F( }' X <w + 5*Q5.1, 1>. x2 d0 x' E# I1 O$ d8 w
]2 L3 [/ ~6 Y W& P7 a# b. `
-8 6 z* n, R* s, V1 E3 K: g! R1 C" K, ?" m# P. C f4 Z
>> FundamentalUnit(Q5) ;0 B4 c. C3 M! \
^ % L+ m6 S _9 w6 m8 c! l1 @! ~Runtime error in 'FundamentalUnit': Field must have positive discriminant 8 @0 |' [) o; h. F2 I) G/ N: T" s8 |% I
8 \* D. `4 s: N2 q. g
>> FundamentalUnit(M); 6 `& ^/ t* P1 P% d) z0 o ^+ E. }+ B5 u3 z; K- P4 ~% }
Runtime error in 'FundamentalUnit': Field must have positive discriminant ; Q3 I2 h% m- s1 Y j; W ; u' Y2 q' @, G% I$ [8+ h$ E/ H: Z8 \0 t4 H! _# ?, G) ~. a
9 s* x, G4 t7 p7 y9 r- y>> Name(M, -50); a) t6 V( O& V0 m/ t ^; ]$ k( ?& {6 Q/ g- r
Runtime error in 'Name': Argument 2 (-50) should be in the range [1 .. 1]! @9 b7 V- A# t; z9 \
2 {# x3 s+ \8 r/ e. X& j+ a1$ F' u5 g7 {2 j) y/ T @
Abelian Group of order 1 ! X/ c5 W U/ I1 nMapping from: Abelian Group of order 1 to Set of ideals of M , F! @, w) o" fAbelian Group of order 1 ! M! r- ?) m c3 `4 PMapping from: Abelian Group of order 1 to Set of ideals of M / U- M3 O! |2 ^" x) _9 V8 {2 r16 m1 n. L I# I, K8 m8 R: v: `$ f
1, j9 x8 u8 ~% M3 a! G4 Z) I3 [- S
Abelian Group of order 1 0 x; P8 C7 R" z5 ]Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no ' e; ?* y6 ?2 T4 t& V B1 r9 N0 _inverse]4 z* x% o" |/ \2 g* }# S" U
1& \, @0 }/ {7 [3 h# m* q$ H$ C# X! [ }
Abelian Group of order 1 ) `. L7 A* u6 w3 R% \Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant / |. S7 L0 t s( I-8 given by a rule [no inverse]# `! G7 I5 o9 |
Abelian Group of order 1 % V7 a/ F/ b8 d8 Q* D- ?2 @Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant0 [1 R! }& o# S5 ^' j
-8 given by a rule [no inverse]. i3 D/ _9 a2 [" p3 m/ U$ C
false7 {3 X! v% Q4 Y8 ~% z
false+ @9 _) R8 w6 s7 p8 i
% p- E+ z7 X: w( X! Y' WQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field4 ]* N5 O, B4 F, R4 U3 U
Univariate Polynomial Ring in w over Q5) M6 Q# C% O0 K2 E
Equation Order of conductor 1 in Q5( K) F: o u# ]- h0 A# E) c- r. q# X+ i
Maximal Equation Order of Q5 + U" }7 {" N3 _: W& P+ ] wQuadratic Field with defining polynomial $.1^2 + 1 over the Rational Field ' j# e5 y: j3 jOrder of conductor 625888888 in Q5# ?2 N& z0 Q# H$ ?' O. Q
true Quadratic Field with defining polynomial $.1^2 + 1 over the Rational Field9 G# C! v6 z+ b
true Maximal Equation Order of Q5 ) r" v8 E0 k8 c" S Z6 Itrue Order of conductor 1 in Q5 % g# X/ E4 G* utrue Order of conductor 1 in Q54 F1 x* E7 \* q. t) U9 t! p4 f- a
true Order of conductor 1 in Q53 E0 ]9 u6 s; o) n I* ~8 W5 ^
[. {: V2 U6 T7 A
<w - Q5.1, 1>, Z! R; a3 {- q6 L
<w + Q5.1, 1> 0 c g- Q2 x8 m$ J$ |) Q] ; Y, a* i' Q8 q$ ~-4; l7 D! w n# y8 O J! S
( O. R& X; ~ B9 C _>> FundamentalUnit(Q5) ;+ H! N* U" @" X
^+ R& a' Q) P% |( `5 d
Runtime error in 'FundamentalUnit': Field must have positive discriminant0 ]9 v# y! U8 h# a% G2 \) H4 b7 u
+ T I- X3 X* y7 D: w
/ \9 Q( C" k5 } e
>> FundamentalUnit(M); * Q4 J/ i% m2 N" f! p, K ^& g8 C/ E' Z, ]
Runtime error in 'FundamentalUnit': Field must have positive discriminant 1 G% o/ K* `6 X& K; Q9 w ! j r9 X" L5 ?2 B43 V; X/ `$ @# J; ^' S7 q
: Y. j! o4 ]/ `
>> Name(M, -1); 9 c0 i& w/ g5 ^% U' g( [2 T# u ^6 |% g4 r5 J5 L) m8 k
Runtime error in 'Name': Argument 2 (-1) should be in the range [1 .. 1]* t$ \( ]* m$ y
" q6 P& O; l7 g1 2 `# j6 ? d b" mAbelian Group of order 1 9 q3 |. S* u5 i$ C$ q P; qMapping from: Abelian Group of order 1 to Set of ideals of M9 z" L1 i, E% X0 U( o0 C( H
Abelian Group of order 1 i. ^1 Q; O5 j4 s
Mapping from: Abelian Group of order 1 to Set of ideals of M7 ]9 }+ g W" z4 l8 }% B
1" v3 O/ j3 \4 w+ w( b- a" K7 p
15 s5 p! H! A3 k/ K- ]) I6 L
Abelian Group of order 19 o9 `$ d4 Y4 l$ @
Mapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no+ d# _- \3 p4 U; w
inverse] 5 L6 J/ D/ v7 I( ^* e0 |14 g3 ?; p8 f: \! t L: @. b
Abelian Group of order 1. O/ @4 ~! { [# V
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant ( w2 o1 A* |. m3 e, I, f-4 given by a rule [no inverse]7 R3 h% ], K2 `2 y1 u+ ^
Abelian Group of order 1 0 R, E8 d6 I, {+ O, ^Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant - H0 z* X9 I& t0 K0 U1 J9 P: P-4 given by a rule [no inverse]+ O$ @2 B$ n6 d
false* k+ s$ |/ Z& R6 w2 b
false a- H( w' x0 M8 |+ C( c( q8 R* m
===============( M" F8 {, r4 c
6 g* R1 b; `% @0 d% l7 ?- C" i: IQ5:=QuadraticField(-3) ;& B0 [% T5 K F* K( p
Q5;: _7 F# }7 d" s+ y& f
6 ^8 D* J$ v1 d6 G$ W
Q<w> :=PolynomialRing(Q5);Q;/ M k9 B) Y+ c, X& Z; g! [7 I
EquationOrder(Q5); , M$ q" L% q$ u) c- OM:=MaximalOrder(Q5) ;0 I( j7 A' z+ H1 h3 [
M;! N: J5 q0 e% w1 j7 R
NumberField(M);. V* E9 R3 v) b* n
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; , S6 ~5 z6 `1 q8 V0 m2 RIsQuadratic(Q5);' Y0 n9 \! M3 t& w7 F6 W+ G* V
IsQuadratic(S1); 0 t9 v" d8 O4 `4 }9 M4 Z9 {3 hIsQuadratic(S4); 5 ^2 m8 k4 D* Z3 a+ w4 \IsQuadratic(S25); , x9 o3 J; ~9 G) d- N% PIsQuadratic(S625888888);5 l }( M. n4 e l6 |" J
Factorization(w^2+3); m" G% {8 S, E3 f6 A DDiscriminant(Q5) ;4 z: ]4 m, Q9 j
FundamentalUnit(Q5) ;& {. E+ j! ~8 L# O- N1 R& p
FundamentalUnit(M);- E7 L$ [# y( m( `5 r+ I
Conductor(Q5) ;4 Q: j# O' }) u( @6 n+ u
# S6 d: @. m5 k$ S: OName(M, -3); . y {5 W; f* K3 uConductor(M); # I& X* Z, e1 _0 ~7 W' g3 v* OClassGroup(Q5) ; 4 y3 q6 Z) f. i' P5 H$ lClassGroup(M); - e1 k# P( M S; d9 `2 X3 yClassNumber(Q5) ; # w3 g8 n; n5 X1 s+ ~/ `6 _ClassNumber(M) ;+ _' R; s4 G. U% i5 n ^( N: M
PicardGroup(M) ;3 R# u4 W N+ q# m) n6 f# d1 ]/ [) E
PicardNumber(M) ; " c7 K3 R" ?5 V3 e" ]5 `( A. L- E
QuadraticClassGroupTwoPart(Q5);' j3 i: E$ ]6 ?" ]; `( j* G
QuadraticClassGroupTwoPart(M);% P% M, C/ @9 z. t" g3 v( p
NormEquation(Q5, -3) ;4 ^. r2 M) D/ I) v& _; s
NormEquation(M, -3) ;& @- u# A$ S" h C6 @7 ~9 S
! s; R. ]6 P. `- v8 F
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field' B7 \! S7 @8 M" n! z# [% s! d
Univariate Polynomial Ring in w over Q58 N* f1 O/ J& z/ A, h, T
Equation Order of conductor 2 in Q5 # a4 n# l* B) B( T' q# q# NMaximal Order of Q5& F3 p( L# d+ D' `, z
Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field; p* @* V0 G$ e+ v! C- ~# H
Order of conductor 625888888 in Q5 0 a6 Z( @2 D; F% o0 h# w1 ^true Quadratic Field with defining polynomial $.1^2 + 3 over the Rational Field 4 x: F8 p' h! b4 U0 {2 Y7 Ltrue Maximal Order of Q5 ) Z) o+ f9 j7 F! Y" j& |7 w8 x3 \6 Utrue Order of conductor 16 in Q5) E8 F$ W; e5 y) q- |
true Order of conductor 625 in Q5 ! V: H8 D! l( Dtrue Order of conductor 391736900121876544 in Q5 7 P- A, J4 X- A( D" C& j, t+ Y* n[ + d$ ~$ T5 t9 _ D! N2 q4 l* A <w - Q5.1, 1>, $ C3 R8 [2 r) _* q! ^ P& y <w + Q5.1, 1>+ V: \6 e4 C# q0 ?+ y
] & h: x5 S- {. O- [! T/ T-3 . g9 ?# z- u& o% x" C' \- U: E; `( O2 \! a
>> FundamentalUnit(Q5) ;& Y/ g& T0 G4 C/ h6 Z8 Q; n7 i
^ ' G$ \/ ` S) r5 Q7 N% }. gRuntime error in 'FundamentalUnit': Field must have positive discriminant5 a/ |( ?$ @: I4 K
8 ?0 |0 G* {* {# s
" l# I- _! G$ {2 Y( v2 X
>> FundamentalUnit(M); 9 t( q' B C6 p7 Y ^) J9 R/ K4 ?% I4 o( ^! v' }
Runtime error in 'FundamentalUnit': Field must have positive discriminant6 I/ g+ T% A2 i! w0 x
, V) L. |# d6 `% { S36 ]- `; ]# X1 }3 ^* L& t' h8 g
& A9 A- e$ T- ^" f& [3 U* f- ~
>> Name(M, -3);" t y1 y" o5 J; q9 D- I q
^ 9 n2 Y. f/ @- ?$ x: Y6 T( bRuntime error in 'Name': Argument 2 (-3) should be in the range [1 .. 1]0 Y2 [8 n' I/ P# U0 J
9 J ?: l/ Y0 ]$ H) K1 6 ?% k( I: i6 p; l# UAbelian Group of order 1; h$ t' Y0 k! G$ M
Mapping from: Abelian Group of order 1 to Set of ideals of M 2 V& T: w8 F% ~4 n7 r1 lAbelian Group of order 1 . [$ c0 r- w* O* s9 k0 n$ NMapping from: Abelian Group of order 1 to Set of ideals of M9 D$ b. z" }1 f$ U% R3 V# f: t
1 4 |, C/ \! q. j% V. b& x1 & w% o- X, E6 K( m1 p& n/ jAbelian Group of order 1 ) p/ J- X6 Q) ?6 _4 kMapping from: Abelian Group of order 1 to Set of ideals of M given by a rule [no2 B) }/ ^* Z% T4 k) Y3 z
inverse]! @2 ~% K% R0 ~
1 7 x5 N5 V5 n- h5 z: }+ I# DAbelian Group of order 19 w4 d2 R5 X2 S+ W) {9 K
Mapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant $ e3 q5 a, g. ^0 x-3 given by a rule [no inverse]* T2 X% ?5 H0 s$ [& z+ {% \
Abelian Group of order 1 ' a$ Q( D- I7 n& i3 y3 e. lMapping from: Abelian Group of order 1 to Binary quadratic forms of discriminant 0 g' r4 u) X; Y" |) G- D- Z-3 given by a rule [no inverse] 0 u/ ?3 M$ i+ [/ w- q% jfalse ! A$ c) ?* O p5 U' Pfalse