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Properties

Label 20.4.288...000.1
Degree $20$
Signature $(4, 8)$
Discriminant $2.889\times 10^{25}$
Root discriminant \(18.75\)
Ramified primes $2,3,5$
Class number $1$
Class group trivial
Galois group $S_5$ (as 20T32)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content 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)
 
Copy content 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)
 
Copy content 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);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(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)
 

\( x^{20} - 5 x^{18} - 2 x^{17} + 5 x^{16} + 8 x^{15} + 2 x^{14} - 16 x^{13} + 21 x^{12} + 28 x^{11} + \cdots - 31 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $20$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(4, 8)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(28888165452349440000000000\) \(\medspace = 2^{36}\cdot 3^{16}\cdot 5^{10}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(18.75\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{13/6}3^{4/5}5^{1/2}\approx 24.1776264427508$
Ramified primes:   \(2\), \(3\), \(5\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q\)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{10}a^{14}+\frac{1}{10}a^{13}-\frac{1}{2}a^{11}+\frac{1}{10}a^{10}+\frac{2}{5}a^{9}+\frac{1}{10}a^{8}+\frac{3}{10}a^{7}+\frac{3}{10}a^{6}-\frac{3}{10}a^{4}+\frac{3}{10}a^{3}-\frac{2}{5}a^{2}+\frac{1}{10}a-\frac{1}{10}$, $\frac{1}{30}a^{15}+\frac{1}{30}a^{14}-\frac{1}{6}a^{13}-\frac{7}{15}a^{11}-\frac{1}{30}a^{10}-\frac{3}{10}a^{9}+\frac{13}{30}a^{8}-\frac{2}{5}a^{7}-\frac{1}{6}a^{6}-\frac{13}{30}a^{5}-\frac{7}{30}a^{4}+\frac{11}{30}a^{3}-\frac{7}{15}a^{2}-\frac{1}{5}a+\frac{1}{6}$, $\frac{1}{30}a^{16}-\frac{2}{15}a^{13}+\frac{1}{30}a^{12}-\frac{1}{15}a^{11}+\frac{13}{30}a^{10}-\frac{7}{15}a^{9}+\frac{11}{30}a^{8}+\frac{1}{3}a^{7}-\frac{1}{6}a^{6}+\frac{1}{5}a^{5}+\frac{4}{15}a^{3}-\frac{1}{30}a^{2}+\frac{1}{15}a+\frac{2}{15}$, $\frac{1}{30}a^{17}-\frac{1}{30}a^{14}+\frac{2}{15}a^{13}-\frac{1}{15}a^{12}-\frac{1}{15}a^{11}-\frac{11}{30}a^{10}-\frac{7}{30}a^{9}+\frac{13}{30}a^{8}+\frac{2}{15}a^{7}-\frac{1}{2}a^{6}-\frac{1}{30}a^{4}+\frac{4}{15}a^{3}-\frac{1}{3}a^{2}+\frac{7}{30}a-\frac{1}{10}$, $\frac{1}{30}a^{18}-\frac{1}{30}a^{14}+\frac{1}{15}a^{13}-\frac{1}{15}a^{12}-\frac{1}{3}a^{11}-\frac{7}{15}a^{10}+\frac{1}{3}a^{9}+\frac{11}{30}a^{8}+\frac{7}{30}a^{6}-\frac{7}{15}a^{5}-\frac{11}{30}a^{4}-\frac{1}{15}a^{3}-\frac{13}{30}a^{2}+\frac{11}{30}$, $\frac{1}{16\cdots 90}a^{19}-\frac{13233815721611}{32\cdots 98}a^{18}+\frac{814429034923}{80\cdots 45}a^{17}-\frac{284275116197}{950655899243970}a^{16}+\frac{19620830308289}{26\cdots 15}a^{15}-\frac{9372384948193}{190131179848794}a^{14}-\frac{585834654182948}{26\cdots 15}a^{13}-\frac{26\cdots 61}{16\cdots 90}a^{12}+\frac{37\cdots 47}{16\cdots 90}a^{11}+\frac{43497074773771}{114618087142890}a^{10}-\frac{626435324286352}{80\cdots 45}a^{9}+\frac{38\cdots 89}{80\cdots 45}a^{8}+\frac{40556800168939}{80\cdots 45}a^{7}+\frac{286985649999458}{26\cdots 15}a^{6}+\frac{265406458596749}{16\cdots 49}a^{5}-\frac{403204547641405}{10\cdots 66}a^{4}+\frac{15\cdots 21}{32\cdots 98}a^{3}+\frac{81072122327407}{475327949621985}a^{2}+\frac{22\cdots 67}{80\cdots 45}a+\frac{23\cdots 87}{16\cdots 90}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  Trivial group, which has order $1$
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  Trivial group, which has order $1$
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $11$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{87399392803888}{80\cdots 45}a^{19}-\frac{25846295206342}{80\cdots 45}a^{18}-\frac{393460344925468}{80\cdots 45}a^{17}-\frac{1540225341763}{95065589924397}a^{16}+\frac{44133658305712}{897841682619305}a^{15}+\frac{42560921693464}{475327949621985}a^{14}+\frac{3299070081568}{26\cdots 15}a^{13}-\frac{256001197752116}{16\cdots 49}a^{12}+\frac{368960127388496}{16\cdots 49}a^{11}+\frac{12815492261366}{57309043571445}a^{10}-\frac{96\cdots 96}{80\cdots 45}a^{9}-\frac{11\cdots 32}{80\cdots 45}a^{8}+\frac{10\cdots 84}{16\cdots 49}a^{7}+\frac{47\cdots 64}{26\cdots 15}a^{6}+\frac{55\cdots 72}{80\cdots 45}a^{5}-\frac{747105639499112}{26\cdots 15}a^{4}+\frac{20\cdots 68}{16\cdots 49}a^{3}+\frac{18\cdots 74}{475327949621985}a^{2}+\frac{30\cdots 92}{80\cdots 45}a-\frac{29\cdots 94}{80\cdots 45}$, $\frac{357842278915121}{80\cdots 45}a^{19}-\frac{279102541250459}{80\cdots 45}a^{18}-\frac{32\cdots 37}{16\cdots 90}a^{17}+\frac{41894497020968}{475327949621985}a^{16}+\frac{869212709092847}{53\cdots 30}a^{15}+\frac{160072596689941}{950655899243970}a^{14}-\frac{28703824839146}{26\cdots 15}a^{13}-\frac{58\cdots 39}{80\cdots 45}a^{12}+\frac{50\cdots 29}{32\cdots 98}a^{11}+\frac{2831780552249}{38206029047630}a^{10}-\frac{46\cdots 02}{80\cdots 45}a^{9}-\frac{25\cdots 03}{16\cdots 90}a^{8}+\frac{13\cdots 61}{32\cdots 98}a^{7}+\frac{14\cdots 37}{53\cdots 30}a^{6}+\frac{35\cdots 93}{16\cdots 90}a^{5}+\frac{169223563148077}{10\cdots 66}a^{4}+\frac{52\cdots 64}{80\cdots 45}a^{3}+\frac{11\cdots 71}{95065589924397}a^{2}+\frac{39\cdots 87}{80\cdots 45}a-\frac{93\cdots 29}{16\cdots 90}$, $\frac{267318394777111}{16\cdots 90}a^{19}-\frac{387719464435471}{16\cdots 90}a^{18}-\frac{434471255792693}{80\cdots 45}a^{17}+\frac{11267879216063}{190131179848794}a^{16}+\frac{4760883930646}{538705009571583}a^{15}+\frac{14097706909313}{190131179848794}a^{14}-\frac{111163261954947}{17\cdots 10}a^{13}-\frac{30\cdots 71}{16\cdots 90}a^{12}+\frac{55\cdots 97}{80\cdots 45}a^{11}-\frac{55320360089561}{114618087142890}a^{10}-\frac{12\cdots 13}{80\cdots 45}a^{9}+\frac{17\cdots 92}{80\cdots 45}a^{8}+\frac{17\cdots 31}{16\cdots 90}a^{7}+\frac{10\cdots 69}{26\cdots 15}a^{6}-\frac{250598054584726}{16\cdots 49}a^{5}+\frac{33\cdots 21}{53\cdots 30}a^{4}+\frac{20\cdots 01}{80\cdots 45}a^{3}+\frac{13\cdots 21}{475327949621985}a^{2}+\frac{13\cdots 33}{16\cdots 90}a+\frac{74\cdots 97}{16\cdots 90}$, $\frac{14842055501986}{475327949621985}a^{19}-\frac{9906933476899}{475327949621985}a^{18}-\frac{73280909362168}{475327949621985}a^{17}+\frac{56545427922289}{950655899243970}a^{16}+\frac{7758575305802}{52814216624665}a^{15}+\frac{55415056989233}{475327949621985}a^{14}-\frac{369095484854}{31688529974799}a^{13}-\frac{265103175793729}{475327949621985}a^{12}+\frac{523175737068592}{475327949621985}a^{11}+\frac{679852709921}{3371120210085}a^{10}-\frac{20\cdots 22}{475327949621985}a^{9}-\frac{13\cdots 43}{950655899243970}a^{8}+\frac{18\cdots 37}{475327949621985}a^{7}+\frac{132298592197672}{52814216624665}a^{6}-\frac{49205866912172}{95065589924397}a^{5}-\frac{4162793695382}{52814216624665}a^{4}+\frac{23\cdots 06}{475327949621985}a^{3}+\frac{41\cdots 89}{475327949621985}a^{2}+\frac{226291444545178}{95065589924397}a-\frac{17\cdots 89}{950655899243970}$, $\frac{67899970172657}{53\cdots 30}a^{19}-\frac{18922171284522}{897841682619305}a^{18}-\frac{93401404272247}{26\cdots 15}a^{17}+\frac{962793611829}{21125686649866}a^{16}+\frac{51445319816297}{26\cdots 15}a^{15}+\frac{2822742918479}{105628433249330}a^{14}-\frac{172243196306847}{17\cdots 10}a^{13}-\frac{76003216877954}{26\cdots 15}a^{12}+\frac{406395052247621}{897841682619305}a^{11}-\frac{11746032728091}{38206029047630}a^{10}-\frac{11\cdots 02}{897841682619305}a^{9}+\frac{506054263368011}{26\cdots 15}a^{8}+\frac{534478461983045}{359136673047722}a^{7}-\frac{33\cdots 23}{53\cdots 30}a^{6}-\frac{20\cdots 71}{26\cdots 15}a^{5}+\frac{60\cdots 91}{53\cdots 30}a^{4}+\frac{24\cdots 31}{897841682619305}a^{3}+\frac{49781702701573}{31688529974799}a^{2}+\frac{739169355816821}{53\cdots 30}a-\frac{672949687907678}{897841682619305}$, $\frac{44553502204631}{16\cdots 90}a^{19}-\frac{26281514640439}{80\cdots 45}a^{18}-\frac{23617675684901}{16\cdots 90}a^{17}-\frac{3337619239313}{475327949621985}a^{16}-\frac{40503421074781}{26\cdots 15}a^{15}+\frac{3748717897796}{95065589924397}a^{14}-\frac{18941515924925}{10\cdots 66}a^{13}+\frac{145683857744483}{80\cdots 45}a^{12}+\frac{367910797703026}{80\cdots 45}a^{11}-\frac{4882043938157}{57309043571445}a^{10}+\frac{792593032182317}{16\cdots 90}a^{9}-\frac{33\cdots 02}{80\cdots 45}a^{8}-\frac{14\cdots 45}{32\cdots 98}a^{7}+\frac{913026874092557}{26\cdots 15}a^{6}+\frac{27\cdots 17}{80\cdots 45}a^{5}-\frac{11125943921482}{538705009571583}a^{4}+\frac{42\cdots 98}{80\cdots 45}a^{3}+\frac{91422682297126}{95065589924397}a^{2}+\frac{83\cdots 07}{80\cdots 45}a+\frac{65\cdots 59}{80\cdots 45}$, $\frac{147654225639251}{80\cdots 45}a^{19}-\frac{1538813972470}{16\cdots 49}a^{18}-\frac{824939818933142}{80\cdots 45}a^{17}-\frac{12719476835093}{950655899243970}a^{16}+\frac{695609959247579}{53\cdots 30}a^{15}+\frac{38858456295191}{475327949621985}a^{14}+\frac{157340895904007}{53\cdots 30}a^{13}-\frac{26\cdots 99}{80\cdots 45}a^{12}+\frac{35\cdots 73}{80\cdots 45}a^{11}+\frac{36221470370782}{57309043571445}a^{10}-\frac{44\cdots 49}{16\cdots 90}a^{9}-\frac{34\cdots 09}{16\cdots 90}a^{8}+\frac{18\cdots 03}{80\cdots 45}a^{7}+\frac{60\cdots 68}{26\cdots 15}a^{6}-\frac{93603390688997}{32\cdots 98}a^{5}-\frac{461510589095839}{26\cdots 15}a^{4}+\frac{51\cdots 59}{16\cdots 90}a^{3}+\frac{31\cdots 53}{475327949621985}a^{2}+\frac{66\cdots 42}{16\cdots 49}a+\frac{46\cdots 01}{16\cdots 90}$, $\frac{157622739492529}{53\cdots 30}a^{19}-\frac{198617271940909}{53\cdots 30}a^{18}-\frac{327201155676431}{26\cdots 15}a^{17}+\frac{3481199305693}{31688529974799}a^{16}+\frac{297384788212829}{26\cdots 15}a^{15}+\frac{6090123800197}{105628433249330}a^{14}-\frac{575169777337823}{53\cdots 30}a^{13}-\frac{23\cdots 23}{53\cdots 30}a^{12}+\frac{625858812441812}{538705009571583}a^{11}-\frac{28571019500669}{114618087142890}a^{10}-\frac{10\cdots 83}{26\cdots 15}a^{9}+\frac{763001920810429}{17\cdots 10}a^{8}+\frac{22\cdots 39}{53\cdots 30}a^{7}+\frac{16\cdots 92}{26\cdots 15}a^{6}-\frac{31\cdots 87}{26\cdots 15}a^{5}-\frac{30\cdots 49}{53\cdots 30}a^{4}+\frac{25\cdots 96}{538705009571583}a^{3}+\frac{185995959584356}{31688529974799}a^{2}-\frac{51\cdots 11}{53\cdots 30}a-\frac{86\cdots 12}{26\cdots 15}$, $\frac{30585415354739}{16\cdots 90}a^{19}+\frac{217110866982169}{16\cdots 90}a^{18}-\frac{736038156720857}{16\cdots 90}a^{17}-\frac{13688123716033}{950655899243970}a^{16}+\frac{150466155790851}{17\cdots 10}a^{15}-\frac{17236110551117}{475327949621985}a^{14}+\frac{649456121441443}{53\cdots 30}a^{13}-\frac{720040687293583}{32\cdots 98}a^{12}+\frac{350718051728903}{16\cdots 90}a^{11}+\frac{12336533169418}{19103014523815}a^{10}-\frac{10\cdots 94}{80\cdots 45}a^{9}-\frac{95\cdots 41}{16\cdots 90}a^{8}+\frac{83\cdots 79}{80\cdots 45}a^{7}+\frac{43\cdots 57}{53\cdots 30}a^{6}+\frac{11\cdots 81}{32\cdots 98}a^{5}-\frac{112602796433819}{179568336523861}a^{4}+\frac{46\cdots 49}{80\cdots 45}a^{3}+\frac{160003652404711}{95065589924397}a^{2}+\frac{58\cdots 33}{16\cdots 90}a-\frac{35\cdots 86}{80\cdots 45}$, $\frac{162951825174464}{80\cdots 45}a^{19}-\frac{117711349929649}{16\cdots 90}a^{18}-\frac{819708026482748}{80\cdots 45}a^{17}+\frac{300924909565}{95065589924397}a^{16}+\frac{598940376806429}{53\cdots 30}a^{15}+\frac{17325234171149}{190131179848794}a^{14}+\frac{49106379392821}{26\cdots 15}a^{13}-\frac{540019350276043}{16\cdots 49}a^{12}+\frac{17\cdots 53}{32\cdots 98}a^{11}+\frac{24782126991083}{57309043571445}a^{10}-\frac{45\cdots 39}{16\cdots 90}a^{9}-\frac{28\cdots 83}{16\cdots 90}a^{8}+\frac{37\cdots 99}{16\cdots 90}a^{7}+\frac{36\cdots 63}{17\cdots 10}a^{6}+\frac{60\cdots 83}{16\cdots 90}a^{5}-\frac{21\cdots 01}{53\cdots 30}a^{4}+\frac{26\cdots 79}{80\cdots 45}a^{3}+\frac{65\cdots 23}{950655899243970}a^{2}+\frac{97\cdots 77}{32\cdots 98}a-\frac{20\cdots 01}{32\cdots 98}$, $\frac{47448326061913}{80\cdots 45}a^{19}+\frac{9070401361651}{16\cdots 90}a^{18}-\frac{400374130166747}{16\cdots 90}a^{17}-\frac{12298871981012}{475327949621985}a^{16}+\frac{164039471210063}{53\cdots 30}a^{15}+\frac{48971472289661}{950655899243970}a^{14}+\frac{3033020320846}{897841682619305}a^{13}-\frac{490993415405291}{16\cdots 90}a^{12}+\frac{676119904563851}{16\cdots 90}a^{11}+\frac{28121176171093}{114618087142890}a^{10}-\frac{49\cdots 96}{80\cdots 45}a^{9}-\frac{19\cdots 39}{16\cdots 90}a^{8}+\frac{49\cdots 61}{16\cdots 90}a^{7}+\frac{135305295298937}{179568336523861}a^{6}+\frac{75\cdots 57}{16\cdots 90}a^{5}+\frac{512262024489143}{17\cdots 10}a^{4}+\frac{94\cdots 41}{80\cdots 45}a^{3}+\frac{12\cdots 71}{475327949621985}a^{2}+\frac{44\cdots 94}{16\cdots 49}a+\frac{15\cdots 98}{80\cdots 45}$ Copy content Toggle raw display
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 110122.225791 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 4 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 110122.225791 \cdot 1}{2\cdot\sqrt{28888165452349440000000000}}\cr\approx \mathstrut & 0.398147445121 \end{aligned}\]

Copy content comment:Analytic class number formula
 
Copy content sage:# 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))))
 
Copy content gp:\\ 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<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); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(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

$S_5$ (as 20T32):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
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.

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content 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.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(2\) Copy content Toggle raw display 2.2.4.16a1.3$x^{8} + 4 x^{7} + 10 x^{6} + 16 x^{5} + 23 x^{4} + 24 x^{3} + 26 x^{2} + 16 x + 11$$4$$2$$16$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
2.2.6.20a1.49$x^{12} + 6 x^{11} + 23 x^{10} + 60 x^{9} + 120 x^{8} + 186 x^{7} + 235 x^{6} + 240 x^{5} + 204 x^{4} + 138 x^{3} + 79 x^{2} + 32 x + 13$$6$$2$$20$$S_4$$$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.2.5.8a1.1$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 720 x^{4} + 640 x^{3} + 400 x^{2} + 160 x + 35$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
3.2.5.8a1.1$x^{10} + 10 x^{9} + 50 x^{8} + 160 x^{7} + 360 x^{6} + 592 x^{5} + 720 x^{4} + 640 x^{3} + 400 x^{2} + 160 x + 35$$5$$2$$8$$F_5$$$[\ ]_{5}^{4}$$
\(5\) Copy content Toggle raw display 5.1.2.1a1.1$x^{2} + 5$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
5.3.2.3a1.2$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 18 x + 14$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)