Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31)
gp: K = bnfinit(y^20 - 5*y^18 - 2*y^17 + 5*y^16 + 8*y^15 + 2*y^14 - 16*y^13 + 21*y^12 + 28*y^11 - 119*y^10 - 150*y^9 + 61*y^8 + 164*y^7 + 62*y^6 - 4*y^5 + 149*y^4 + 396*y^3 + 327*y^2 + 58*y - 31, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31)
\( x^{20} - 5 x^{18} - 2 x^{17} + 5 x^{16} + 8 x^{15} + 2 x^{14} - 16 x^{13} + 21 x^{12} + 28 x^{11} + \cdots - 31 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
oscar: defining_polynomial(K)
Invariants
| Degree: | $20$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
| Signature: | $[4, 8]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
| Discriminant: |
\(28888165452349440000000000\)
\(\medspace = 2^{36}\cdot 3^{16}\cdot 5^{10}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(18.75\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
| Galois root discriminant: | $2^{13/6}3^{4/5}5^{1/2}\approx 24.1776264427508$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{10}a^{14}+\frac{1}{10}a^{13}-\frac{1}{2}a^{11}+\frac{1}{10}a^{10}+\frac{2}{5}a^{9}+\frac{1}{10}a^{8}+\frac{3}{10}a^{7}+\frac{3}{10}a^{6}-\frac{3}{10}a^{4}+\frac{3}{10}a^{3}-\frac{2}{5}a^{2}+\frac{1}{10}a-\frac{1}{10}$, $\frac{1}{30}a^{15}+\frac{1}{30}a^{14}-\frac{1}{6}a^{13}-\frac{7}{15}a^{11}-\frac{1}{30}a^{10}-\frac{3}{10}a^{9}+\frac{13}{30}a^{8}-\frac{2}{5}a^{7}-\frac{1}{6}a^{6}-\frac{13}{30}a^{5}-\frac{7}{30}a^{4}+\frac{11}{30}a^{3}-\frac{7}{15}a^{2}-\frac{1}{5}a+\frac{1}{6}$, $\frac{1}{30}a^{16}-\frac{2}{15}a^{13}+\frac{1}{30}a^{12}-\frac{1}{15}a^{11}+\frac{13}{30}a^{10}-\frac{7}{15}a^{9}+\frac{11}{30}a^{8}+\frac{1}{3}a^{7}-\frac{1}{6}a^{6}+\frac{1}{5}a^{5}+\frac{4}{15}a^{3}-\frac{1}{30}a^{2}+\frac{1}{15}a+\frac{2}{15}$, $\frac{1}{30}a^{17}-\frac{1}{30}a^{14}+\frac{2}{15}a^{13}-\frac{1}{15}a^{12}-\frac{1}{15}a^{11}-\frac{11}{30}a^{10}-\frac{7}{30}a^{9}+\frac{13}{30}a^{8}+\frac{2}{15}a^{7}-\frac{1}{2}a^{6}-\frac{1}{30}a^{4}+\frac{4}{15}a^{3}-\frac{1}{3}a^{2}+\frac{7}{30}a-\frac{1}{10}$, $\frac{1}{30}a^{18}-\frac{1}{30}a^{14}+\frac{1}{15}a^{13}-\frac{1}{15}a^{12}-\frac{1}{3}a^{11}-\frac{7}{15}a^{10}+\frac{1}{3}a^{9}+\frac{11}{30}a^{8}+\frac{7}{30}a^{6}-\frac{7}{15}a^{5}-\frac{11}{30}a^{4}-\frac{1}{15}a^{3}-\frac{13}{30}a^{2}+\frac{11}{30}$, $\frac{1}{16\!\cdots\!90}a^{19}-\frac{13233815721611}{32\!\cdots\!98}a^{18}+\frac{814429034923}{80\!\cdots\!45}a^{17}-\frac{284275116197}{950655899243970}a^{16}+\frac{19620830308289}{26\!\cdots\!15}a^{15}-\frac{9372384948193}{190131179848794}a^{14}-\frac{585834654182948}{26\!\cdots\!15}a^{13}-\frac{26\!\cdots\!61}{16\!\cdots\!90}a^{12}+\frac{37\!\cdots\!47}{16\!\cdots\!90}a^{11}+\frac{43497074773771}{114618087142890}a^{10}-\frac{626435324286352}{80\!\cdots\!45}a^{9}+\frac{38\!\cdots\!89}{80\!\cdots\!45}a^{8}+\frac{40556800168939}{80\!\cdots\!45}a^{7}+\frac{286985649999458}{26\!\cdots\!15}a^{6}+\frac{265406458596749}{16\!\cdots\!49}a^{5}-\frac{403204547641405}{10\!\cdots\!66}a^{4}+\frac{15\!\cdots\!21}{32\!\cdots\!98}a^{3}+\frac{81072122327407}{475327949621985}a^{2}+\frac{22\!\cdots\!67}{80\!\cdots\!45}a+\frac{23\!\cdots\!87}{16\!\cdots\!90}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
oscar: class_group(K)
Unit group
sage: UK = K.unit_group()
magma: UK, fUK := UnitGroup(K);
oscar: UK, fUK = unit_group(OK)
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -1 \)
(order $2$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
| Fundamental units: |
$\frac{87399392803888}{80\!\cdots\!45}a^{19}-\frac{25846295206342}{80\!\cdots\!45}a^{18}-\frac{393460344925468}{80\!\cdots\!45}a^{17}-\frac{1540225341763}{95065589924397}a^{16}+\frac{44133658305712}{897841682619305}a^{15}+\frac{42560921693464}{475327949621985}a^{14}+\frac{3299070081568}{26\!\cdots\!15}a^{13}-\frac{256001197752116}{16\!\cdots\!49}a^{12}+\frac{368960127388496}{16\!\cdots\!49}a^{11}+\frac{12815492261366}{57309043571445}a^{10}-\frac{96\!\cdots\!96}{80\!\cdots\!45}a^{9}-\frac{11\!\cdots\!32}{80\!\cdots\!45}a^{8}+\frac{10\!\cdots\!84}{16\!\cdots\!49}a^{7}+\frac{47\!\cdots\!64}{26\!\cdots\!15}a^{6}+\frac{55\!\cdots\!72}{80\!\cdots\!45}a^{5}-\frac{747105639499112}{26\!\cdots\!15}a^{4}+\frac{20\!\cdots\!68}{16\!\cdots\!49}a^{3}+\frac{18\!\cdots\!74}{475327949621985}a^{2}+\frac{30\!\cdots\!92}{80\!\cdots\!45}a-\frac{29\!\cdots\!94}{80\!\cdots\!45}$, $\frac{357842278915121}{80\!\cdots\!45}a^{19}-\frac{279102541250459}{80\!\cdots\!45}a^{18}-\frac{32\!\cdots\!37}{16\!\cdots\!90}a^{17}+\frac{41894497020968}{475327949621985}a^{16}+\frac{869212709092847}{53\!\cdots\!30}a^{15}+\frac{160072596689941}{950655899243970}a^{14}-\frac{28703824839146}{26\!\cdots\!15}a^{13}-\frac{58\!\cdots\!39}{80\!\cdots\!45}a^{12}+\frac{50\!\cdots\!29}{32\!\cdots\!98}a^{11}+\frac{2831780552249}{38206029047630}a^{10}-\frac{46\!\cdots\!02}{80\!\cdots\!45}a^{9}-\frac{25\!\cdots\!03}{16\!\cdots\!90}a^{8}+\frac{13\!\cdots\!61}{32\!\cdots\!98}a^{7}+\frac{14\!\cdots\!37}{53\!\cdots\!30}a^{6}+\frac{35\!\cdots\!93}{16\!\cdots\!90}a^{5}+\frac{169223563148077}{10\!\cdots\!66}a^{4}+\frac{52\!\cdots\!64}{80\!\cdots\!45}a^{3}+\frac{11\!\cdots\!71}{95065589924397}a^{2}+\frac{39\!\cdots\!87}{80\!\cdots\!45}a-\frac{93\!\cdots\!29}{16\!\cdots\!90}$, $\frac{267318394777111}{16\!\cdots\!90}a^{19}-\frac{387719464435471}{16\!\cdots\!90}a^{18}-\frac{434471255792693}{80\!\cdots\!45}a^{17}+\frac{11267879216063}{190131179848794}a^{16}+\frac{4760883930646}{538705009571583}a^{15}+\frac{14097706909313}{190131179848794}a^{14}-\frac{111163261954947}{17\!\cdots\!10}a^{13}-\frac{30\!\cdots\!71}{16\!\cdots\!90}a^{12}+\frac{55\!\cdots\!97}{80\!\cdots\!45}a^{11}-\frac{55320360089561}{114618087142890}a^{10}-\frac{12\!\cdots\!13}{80\!\cdots\!45}a^{9}+\frac{17\!\cdots\!92}{80\!\cdots\!45}a^{8}+\frac{17\!\cdots\!31}{16\!\cdots\!90}a^{7}+\frac{10\!\cdots\!69}{26\!\cdots\!15}a^{6}-\frac{250598054584726}{16\!\cdots\!49}a^{5}+\frac{33\!\cdots\!21}{53\!\cdots\!30}a^{4}+\frac{20\!\cdots\!01}{80\!\cdots\!45}a^{3}+\frac{13\!\cdots\!21}{475327949621985}a^{2}+\frac{13\!\cdots\!33}{16\!\cdots\!90}a+\frac{74\!\cdots\!97}{16\!\cdots\!90}$, $\frac{14842055501986}{475327949621985}a^{19}-\frac{9906933476899}{475327949621985}a^{18}-\frac{73280909362168}{475327949621985}a^{17}+\frac{56545427922289}{950655899243970}a^{16}+\frac{7758575305802}{52814216624665}a^{15}+\frac{55415056989233}{475327949621985}a^{14}-\frac{369095484854}{31688529974799}a^{13}-\frac{265103175793729}{475327949621985}a^{12}+\frac{523175737068592}{475327949621985}a^{11}+\frac{679852709921}{3371120210085}a^{10}-\frac{20\!\cdots\!22}{475327949621985}a^{9}-\frac{13\!\cdots\!43}{950655899243970}a^{8}+\frac{18\!\cdots\!37}{475327949621985}a^{7}+\frac{132298592197672}{52814216624665}a^{6}-\frac{49205866912172}{95065589924397}a^{5}-\frac{4162793695382}{52814216624665}a^{4}+\frac{23\!\cdots\!06}{475327949621985}a^{3}+\frac{41\!\cdots\!89}{475327949621985}a^{2}+\frac{226291444545178}{95065589924397}a-\frac{17\!\cdots\!89}{950655899243970}$, $\frac{67899970172657}{53\!\cdots\!30}a^{19}-\frac{18922171284522}{897841682619305}a^{18}-\frac{93401404272247}{26\!\cdots\!15}a^{17}+\frac{962793611829}{21125686649866}a^{16}+\frac{51445319816297}{26\!\cdots\!15}a^{15}+\frac{2822742918479}{105628433249330}a^{14}-\frac{172243196306847}{17\!\cdots\!10}a^{13}-\frac{76003216877954}{26\!\cdots\!15}a^{12}+\frac{406395052247621}{897841682619305}a^{11}-\frac{11746032728091}{38206029047630}a^{10}-\frac{11\!\cdots\!02}{897841682619305}a^{9}+\frac{506054263368011}{26\!\cdots\!15}a^{8}+\frac{534478461983045}{359136673047722}a^{7}-\frac{33\!\cdots\!23}{53\!\cdots\!30}a^{6}-\frac{20\!\cdots\!71}{26\!\cdots\!15}a^{5}+\frac{60\!\cdots\!91}{53\!\cdots\!30}a^{4}+\frac{24\!\cdots\!31}{897841682619305}a^{3}+\frac{49781702701573}{31688529974799}a^{2}+\frac{739169355816821}{53\!\cdots\!30}a-\frac{672949687907678}{897841682619305}$, $\frac{44553502204631}{16\!\cdots\!90}a^{19}-\frac{26281514640439}{80\!\cdots\!45}a^{18}-\frac{23617675684901}{16\!\cdots\!90}a^{17}-\frac{3337619239313}{475327949621985}a^{16}-\frac{40503421074781}{26\!\cdots\!15}a^{15}+\frac{3748717897796}{95065589924397}a^{14}-\frac{18941515924925}{10\!\cdots\!66}a^{13}+\frac{145683857744483}{80\!\cdots\!45}a^{12}+\frac{367910797703026}{80\!\cdots\!45}a^{11}-\frac{4882043938157}{57309043571445}a^{10}+\frac{792593032182317}{16\!\cdots\!90}a^{9}-\frac{33\!\cdots\!02}{80\!\cdots\!45}a^{8}-\frac{14\!\cdots\!45}{32\!\cdots\!98}a^{7}+\frac{913026874092557}{26\!\cdots\!15}a^{6}+\frac{27\!\cdots\!17}{80\!\cdots\!45}a^{5}-\frac{11125943921482}{538705009571583}a^{4}+\frac{42\!\cdots\!98}{80\!\cdots\!45}a^{3}+\frac{91422682297126}{95065589924397}a^{2}+\frac{83\!\cdots\!07}{80\!\cdots\!45}a+\frac{65\!\cdots\!59}{80\!\cdots\!45}$, $\frac{147654225639251}{80\!\cdots\!45}a^{19}-\frac{1538813972470}{16\!\cdots\!49}a^{18}-\frac{824939818933142}{80\!\cdots\!45}a^{17}-\frac{12719476835093}{950655899243970}a^{16}+\frac{695609959247579}{53\!\cdots\!30}a^{15}+\frac{38858456295191}{475327949621985}a^{14}+\frac{157340895904007}{53\!\cdots\!30}a^{13}-\frac{26\!\cdots\!99}{80\!\cdots\!45}a^{12}+\frac{35\!\cdots\!73}{80\!\cdots\!45}a^{11}+\frac{36221470370782}{57309043571445}a^{10}-\frac{44\!\cdots\!49}{16\!\cdots\!90}a^{9}-\frac{34\!\cdots\!09}{16\!\cdots\!90}a^{8}+\frac{18\!\cdots\!03}{80\!\cdots\!45}a^{7}+\frac{60\!\cdots\!68}{26\!\cdots\!15}a^{6}-\frac{93603390688997}{32\!\cdots\!98}a^{5}-\frac{461510589095839}{26\!\cdots\!15}a^{4}+\frac{51\!\cdots\!59}{16\!\cdots\!90}a^{3}+\frac{31\!\cdots\!53}{475327949621985}a^{2}+\frac{66\!\cdots\!42}{16\!\cdots\!49}a+\frac{46\!\cdots\!01}{16\!\cdots\!90}$, $\frac{157622739492529}{53\!\cdots\!30}a^{19}-\frac{198617271940909}{53\!\cdots\!30}a^{18}-\frac{327201155676431}{26\!\cdots\!15}a^{17}+\frac{3481199305693}{31688529974799}a^{16}+\frac{297384788212829}{26\!\cdots\!15}a^{15}+\frac{6090123800197}{105628433249330}a^{14}-\frac{575169777337823}{53\!\cdots\!30}a^{13}-\frac{23\!\cdots\!23}{53\!\cdots\!30}a^{12}+\frac{625858812441812}{538705009571583}a^{11}-\frac{28571019500669}{114618087142890}a^{10}-\frac{10\!\cdots\!83}{26\!\cdots\!15}a^{9}+\frac{763001920810429}{17\!\cdots\!10}a^{8}+\frac{22\!\cdots\!39}{53\!\cdots\!30}a^{7}+\frac{16\!\cdots\!92}{26\!\cdots\!15}a^{6}-\frac{31\!\cdots\!87}{26\!\cdots\!15}a^{5}-\frac{30\!\cdots\!49}{53\!\cdots\!30}a^{4}+\frac{25\!\cdots\!96}{538705009571583}a^{3}+\frac{185995959584356}{31688529974799}a^{2}-\frac{51\!\cdots\!11}{53\!\cdots\!30}a-\frac{86\!\cdots\!12}{26\!\cdots\!15}$, $\frac{30585415354739}{16\!\cdots\!90}a^{19}+\frac{217110866982169}{16\!\cdots\!90}a^{18}-\frac{736038156720857}{16\!\cdots\!90}a^{17}-\frac{13688123716033}{950655899243970}a^{16}+\frac{150466155790851}{17\!\cdots\!10}a^{15}-\frac{17236110551117}{475327949621985}a^{14}+\frac{649456121441443}{53\!\cdots\!30}a^{13}-\frac{720040687293583}{32\!\cdots\!98}a^{12}+\frac{350718051728903}{16\!\cdots\!90}a^{11}+\frac{12336533169418}{19103014523815}a^{10}-\frac{10\!\cdots\!94}{80\!\cdots\!45}a^{9}-\frac{95\!\cdots\!41}{16\!\cdots\!90}a^{8}+\frac{83\!\cdots\!79}{80\!\cdots\!45}a^{7}+\frac{43\!\cdots\!57}{53\!\cdots\!30}a^{6}+\frac{11\!\cdots\!81}{32\!\cdots\!98}a^{5}-\frac{112602796433819}{179568336523861}a^{4}+\frac{46\!\cdots\!49}{80\!\cdots\!45}a^{3}+\frac{160003652404711}{95065589924397}a^{2}+\frac{58\!\cdots\!33}{16\!\cdots\!90}a-\frac{35\!\cdots\!86}{80\!\cdots\!45}$, $\frac{162951825174464}{80\!\cdots\!45}a^{19}-\frac{117711349929649}{16\!\cdots\!90}a^{18}-\frac{819708026482748}{80\!\cdots\!45}a^{17}+\frac{300924909565}{95065589924397}a^{16}+\frac{598940376806429}{53\!\cdots\!30}a^{15}+\frac{17325234171149}{190131179848794}a^{14}+\frac{49106379392821}{26\!\cdots\!15}a^{13}-\frac{540019350276043}{16\!\cdots\!49}a^{12}+\frac{17\!\cdots\!53}{32\!\cdots\!98}a^{11}+\frac{24782126991083}{57309043571445}a^{10}-\frac{45\!\cdots\!39}{16\!\cdots\!90}a^{9}-\frac{28\!\cdots\!83}{16\!\cdots\!90}a^{8}+\frac{37\!\cdots\!99}{16\!\cdots\!90}a^{7}+\frac{36\!\cdots\!63}{17\!\cdots\!10}a^{6}+\frac{60\!\cdots\!83}{16\!\cdots\!90}a^{5}-\frac{21\!\cdots\!01}{53\!\cdots\!30}a^{4}+\frac{26\!\cdots\!79}{80\!\cdots\!45}a^{3}+\frac{65\!\cdots\!23}{950655899243970}a^{2}+\frac{97\!\cdots\!77}{32\!\cdots\!98}a-\frac{20\!\cdots\!01}{32\!\cdots\!98}$, $\frac{47448326061913}{80\!\cdots\!45}a^{19}+\frac{9070401361651}{16\!\cdots\!90}a^{18}-\frac{400374130166747}{16\!\cdots\!90}a^{17}-\frac{12298871981012}{475327949621985}a^{16}+\frac{164039471210063}{53\!\cdots\!30}a^{15}+\frac{48971472289661}{950655899243970}a^{14}+\frac{3033020320846}{897841682619305}a^{13}-\frac{490993415405291}{16\!\cdots\!90}a^{12}+\frac{676119904563851}{16\!\cdots\!90}a^{11}+\frac{28121176171093}{114618087142890}a^{10}-\frac{49\!\cdots\!96}{80\!\cdots\!45}a^{9}-\frac{19\!\cdots\!39}{16\!\cdots\!90}a^{8}+\frac{49\!\cdots\!61}{16\!\cdots\!90}a^{7}+\frac{135305295298937}{179568336523861}a^{6}+\frac{75\!\cdots\!57}{16\!\cdots\!90}a^{5}+\frac{512262024489143}{17\!\cdots\!10}a^{4}+\frac{94\!\cdots\!41}{80\!\cdots\!45}a^{3}+\frac{12\!\cdots\!71}{475327949621985}a^{2}+\frac{44\!\cdots\!94}{16\!\cdots\!49}a+\frac{15\!\cdots\!98}{80\!\cdots\!45}$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
| Regulator: | \( 110122.225791 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 110122.225791 \cdot 1}{2\cdot\sqrt{28888165452349440000000000}}\cr\approx \mathstrut & 0.398147445121 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31)
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
# self-contained Pari/GP code snippet to compute the analytic class number formula
K = bnfinit(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31, 1);
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
/* self-contained Magma code snippet to compute the analytic class number formula */
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31);
OK := Integers(K); DK := Discriminant(OK);
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
hK := #clK; wK := #TorsionSubgroup(UK);
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
# self-contained Oscar code snippet to compute the analytic class number formula
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^20 - 5*x^18 - 2*x^17 + 5*x^16 + 8*x^15 + 2*x^14 - 16*x^13 + 21*x^12 + 28*x^11 - 119*x^10 - 150*x^9 + 61*x^8 + 164*x^7 + 62*x^6 - 4*x^5 + 149*x^4 + 396*x^3 + 327*x^2 + 58*x - 31);
OK = ring_of_integers(K); DK = discriminant(OK);
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
hK = order(clK); wK = torsion_units_order(K);
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: G = GaloisGroup(K);
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
| A non-solvable group of order 120 |
| The 7 conjugacy class representatives for $S_5$ |
| Character table for $S_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.2.214990848000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
sage: K.subfields()[1:-1]
gp: L = nfsubfields(K); L[2..length(b)]
magma: L := Subfields(K); L[2..#L];
oscar: subfields(K)[2:end-1]
Sibling fields
| Degree 5 sibling: | 5.1.103680.1 |
| Degree 6 sibling: | 6.2.10368000.1 |
| Degree 10 siblings: | 10.2.214990848000.1, 10.2.1343692800000.3 |
| Degree 12 sibling: | 12.4.107495424000000.1 |
| Degree 15 sibling: | 15.3.2229025112064000000.1 |
| Degree 20 siblings: | 20.0.369768517790072832000000000.1, deg 20 |
| Degree 24 sibling: | data not computed |
| Degree 30 siblings: | data not computed |
| Degree 40 sibling: | data not computed |
| Minimal sibling: | 5.1.103680.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | R | R | ${\href{/padicField/7.6.0.1}{6} }^{3}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }^{3}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.3.0.1}{3} }^{6}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.8.16.9 | $x^{8} + 4 x^{7} + 18 x^{6} + 48 x^{5} + 103 x^{4} + 144 x^{3} + 138 x^{2} + 128 x + 67$ | $4$ | $2$ | $16$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ |
| 2.12.20.34 | $x^{12} + 4 x^{11} + 12 x^{10} + 24 x^{9} + 44 x^{8} + 32 x^{7} + 48 x^{6} + 16 x^{5} + 68 x^{4} + 32 x^{3} + 24 x^{2} + 28$ | $6$ | $2$ | $20$ | $S_4$ | $[8/3, 8/3]_{3}^{2}$ | |
|
\(3\)
| 3.10.8.1 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ |
| 3.10.8.1 | $x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 598 x^{5} + 750 x^{4} + 640 x^{3} + 280 x^{2} + 40 x + 17$ | $5$ | $2$ | $8$ | $F_5$ | $[\ ]_{5}^{4}$ | |
|
\(5\)
| 5.2.1.1 | $x^{2} + 5$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.1 | $x^{6} + 60 x^{5} + 1221 x^{4} + 8846 x^{3} + 9864 x^{2} + 29208 x + 29309$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |