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Properties

Label 20.16.305...000.1
Degree $20$
Signature $[16, 2]$
Discriminant $3.057\times 10^{30}$
Root discriminant \(33.44\)
Ramified primes $2,5,13,29,31$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^8.\POPlus(4,5)$ (as 20T1013)

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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 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1)
 
Copy content gp:K = bnfinit(y^20 - 18*y^18 + 93*y^16 + 44*y^14 - 1686*y^12 + 4800*y^10 - 4846*y^8 + 1428*y^6 - 59*y^4 - 14*y^2 + 1, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1)
 

\( x^{20} - 18 x^{18} + 93 x^{16} + 44 x^{14} - 1686 x^{12} + 4800 x^{10} - 4846 x^{8} + 1428 x^{6} + \cdots + 1 \) 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:  $[16, 2]$
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:   \(3056554299829282570240000000000\) \(\medspace = 2^{24}\cdot 5^{10}\cdot 13^{4}\cdot 29^{4}\cdot 31^{4}\) 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:  \(33.44\)
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:  not computed
Ramified primes:   \(2\), \(5\), \(13\), \(29\), \(31\) 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:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$, $\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a+\frac{3}{8}$, $\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{8}a^{3}-\frac{1}{4}a^{2}+\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{8}-\frac{1}{8}a^{4}+\frac{1}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{9}-\frac{1}{8}a^{5}+\frac{1}{8}a$, $\frac{1}{16}a^{14}-\frac{1}{16}a^{12}-\frac{1}{16}a^{10}+\frac{1}{16}a^{8}+\frac{3}{16}a^{6}-\frac{3}{16}a^{4}-\frac{1}{2}a^{3}+\frac{5}{16}a^{2}-\frac{1}{2}a-\frac{5}{16}$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{1}{16}a^{11}+\frac{1}{16}a^{9}+\frac{3}{16}a^{7}-\frac{3}{16}a^{5}+\frac{5}{16}a^{3}-\frac{1}{2}a^{2}-\frac{5}{16}a-\frac{1}{2}$, $\frac{1}{16}a^{16}-\frac{1}{8}a^{8}+\frac{1}{16}$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{16}-\frac{1}{16}a^{9}+\frac{1}{16}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{7}{32}a-\frac{9}{32}$, $\frac{1}{90304}a^{18}-\frac{2613}{90304}a^{16}+\frac{149}{5644}a^{14}-\frac{9}{1328}a^{12}-\frac{2577}{45152}a^{10}-\frac{4069}{45152}a^{8}+\frac{147}{2822}a^{6}-\frac{3603}{22576}a^{4}-\frac{1}{2}a^{3}+\frac{24513}{90304}a^{2}-\frac{1}{2}a+\frac{30495}{90304}$, $\frac{1}{180608}a^{19}-\frac{1}{180608}a^{18}-\frac{2613}{180608}a^{17}+\frac{2613}{180608}a^{16}-\frac{815}{45152}a^{15}+\frac{815}{45152}a^{14}-\frac{23}{664}a^{13}+\frac{23}{664}a^{12}+\frac{245}{90304}a^{11}-\frac{245}{90304}a^{10}+\frac{10041}{90304}a^{9}-\frac{10041}{90304}a^{8}-\frac{3057}{45152}a^{7}+\frac{3057}{45152}a^{6}+\frac{863}{11288}a^{5}-\frac{863}{11288}a^{4}-\frac{3707}{180608}a^{3}+\frac{3707}{180608}a^{2}-\frac{65453}{180608}a+\frac{65453}{180608}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  Trivial group, which has order $1$ (assuming GRH)
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:  $C_{2}$, which has order $2$ (assuming GRH)
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:  $17$
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:   $\frac{536649}{90304}a^{18}-\frac{9606837}{90304}a^{16}+\frac{12240571}{22576}a^{14}+\frac{209083}{664}a^{12}-\frac{450998727}{45152}a^{10}+\frac{1243544653}{45152}a^{8}-\frac{588886651}{22576}a^{6}+\frac{66755401}{11288}a^{4}+\frac{21023669}{90304}a^{2}-\frac{5293813}{90304}$, $\frac{536649}{90304}a^{19}-\frac{9606837}{90304}a^{17}+\frac{12240571}{22576}a^{15}+\frac{209083}{664}a^{13}-\frac{450998727}{45152}a^{11}+\frac{1243544653}{45152}a^{9}-\frac{588886651}{22576}a^{7}+\frac{66755401}{11288}a^{5}+\frac{21023669}{90304}a^{3}-\frac{5293813}{90304}a$, $\frac{186307}{10624}a^{19}-\frac{997587}{180608}a^{18}-\frac{3334899}{10624}a^{17}+\frac{17854115}{180608}a^{16}+\frac{4248267}{2656}a^{15}-\frac{22735759}{45152}a^{14}+\frac{618591}{664}a^{13}-\frac{97825}{332}a^{12}-\frac{156564033}{5312}a^{11}+\frac{838200689}{90304}a^{10}+\frac{431478959}{5312}a^{9}-\frac{2308212799}{90304}a^{8}-\frac{204119387}{2656}a^{7}+\frac{1090416095}{45152}a^{6}+\frac{11516737}{664}a^{5}-\frac{30589151}{5644}a^{4}+\frac{7468303}{10624}a^{3}-\frac{41433631}{180608}a^{2}-\frac{1850939}{10624}a+\frac{9824619}{180608}$, $\frac{173045}{22576}a^{19}+\frac{140917}{90304}a^{18}-\frac{6196395}{45152}a^{17}-\frac{2521607}{90304}a^{16}+\frac{15795711}{22576}a^{15}+\frac{802401}{5644}a^{14}+\frac{537121}{1328}a^{13}+\frac{111207}{1328}a^{12}-\frac{290879645}{22576}a^{11}-\frac{118402337}{45152}a^{10}+\frac{401341955}{11288}a^{9}+\frac{325661241}{45152}a^{8}-\frac{761463891}{22576}a^{7}-\frac{9587791}{1411}a^{6}+\frac{173922299}{22576}a^{5}+\frac{33936373}{22576}a^{4}+\frac{1627809}{5644}a^{3}+\frac{6096621}{90304}a^{2}-\frac{3568601}{45152}a-\frac{1277035}{90304}$, $\frac{3243205}{180608}a^{19}+\frac{1088829}{180608}a^{18}-\frac{58048985}{180608}a^{17}-\frac{19486729}{180608}a^{16}+\frac{73933693}{45152}a^{15}+\frac{24813043}{45152}a^{14}+\frac{634829}{664}a^{13}+\frac{427583}{1328}a^{12}-\frac{2725272799}{90304}a^{11}-\frac{914915531}{90304}a^{10}+\frac{7507501349}{90304}a^{9}+\frac{2518828569}{90304}a^{8}-\frac{3548700389}{45152}a^{7}-\frac{1189025555}{45152}a^{6}+\frac{199510673}{11288}a^{5}+\frac{132863481}{22576}a^{4}+\frac{133102473}{180608}a^{3}+\frac{46095113}{180608}a^{2}-\frac{32348353}{180608}a-\frac{10631193}{180608}$, $\frac{613203}{90304}a^{19}-\frac{276005}{90304}a^{18}-\frac{10976461}{90304}a^{17}+\frac{4940779}{90304}a^{16}+\frac{3495811}{5644}a^{15}-\frac{3147419}{11288}a^{14}+\frac{478753}{1328}a^{13}-\frac{215281}{1328}a^{12}-\frac{515291635}{45152}a^{11}+\frac{231953829}{45152}a^{10}+\frac{1420303223}{45152}a^{9}-\frac{639441153}{45152}a^{8}-\frac{42011561}{1411}a^{7}+\frac{151339085}{11288}a^{6}+\frac{152084835}{22576}a^{5}-\frac{68485683}{22576}a^{4}+\frac{24274899}{90304}a^{3}-\frac{11061669}{90304}a^{2}-\frac{6433281}{90304}a+\frac{2864887}{90304}$, $\frac{1631079}{180608}a^{19}-\frac{613941}{180608}a^{18}-\frac{29195679}{180608}a^{17}+\frac{10988717}{180608}a^{16}+\frac{37190251}{45152}a^{15}-\frac{13995559}{45152}a^{14}+\frac{159329}{332}a^{13}-\frac{240413}{1328}a^{12}-\frac{1370592381}{90304}a^{11}+\frac{515909451}{90304}a^{10}+\frac{3777101243}{90304}a^{9}-\frac{1421104357}{90304}a^{8}-\frac{1787048203}{45152}a^{7}+\frac{671595055}{45152}a^{6}+\frac{50505595}{5644}a^{5}-\frac{75499543}{22576}a^{4}+\frac{63242723}{180608}a^{3}-\frac{24242609}{180608}a^{2}-\frac{15808359}{180608}a+\frac{6027053}{180608}$, $\frac{241013}{22576}a^{19}+\frac{35005}{5312}a^{18}-\frac{2157205}{11288}a^{17}-\frac{626557}{5312}a^{16}+\frac{5496971}{5644}a^{15}+\frac{399031}{664}a^{14}+\frac{375751}{664}a^{13}+\frac{465523}{1328}a^{12}-\frac{202537105}{11288}a^{11}-\frac{29414701}{2656}a^{10}+\frac{558419971}{11288}a^{9}+\frac{81045143}{2656}a^{8}-\frac{66110770}{1411}a^{7}-\frac{19162523}{664}a^{6}+\frac{120033319}{11288}a^{5}+\frac{8636781}{1328}a^{4}+\frac{9209729}{22576}a^{3}+\frac{1420621}{5312}a^{2}-\frac{304089}{2822}a-\frac{352385}{5312}$, $\frac{3243205}{180608}a^{19}-\frac{1088829}{180608}a^{18}-\frac{58048985}{180608}a^{17}+\frac{19486729}{180608}a^{16}+\frac{73933693}{45152}a^{15}-\frac{24813043}{45152}a^{14}+\frac{634829}{664}a^{13}-\frac{427583}{1328}a^{12}-\frac{2725272799}{90304}a^{11}+\frac{914915531}{90304}a^{10}+\frac{7507501349}{90304}a^{9}-\frac{2518828569}{90304}a^{8}-\frac{3548700389}{45152}a^{7}+\frac{1189025555}{45152}a^{6}+\frac{199510673}{11288}a^{5}-\frac{132863481}{22576}a^{4}+\frac{133102473}{180608}a^{3}-\frac{46095113}{180608}a^{2}-\frac{32348353}{180608}a+\frac{10631193}{180608}$, $\frac{586795}{22576}a^{19}+\frac{771661}{90304}a^{18}-\frac{21007843}{45152}a^{17}-\frac{13811019}{90304}a^{16}+\frac{53526547}{22576}a^{15}+\frac{17588265}{22576}a^{14}+\frac{1832267}{1328}a^{13}+\frac{151245}{332}a^{12}-\frac{986252111}{22576}a^{11}-\frac{648394131}{45152}a^{10}+\frac{1359227535}{11288}a^{9}+\frac{1785726139}{45152}a^{8}-\frac{2572801059}{22576}a^{7}-\frac{843734285}{22576}a^{6}+\frac{581292125}{22576}a^{5}+\frac{23692515}{2822}a^{4}+\frac{5901693}{5644}a^{3}+\frac{31178649}{90304}a^{2}-\frac{11672729}{45152}a-\frac{7620139}{90304}$, $\frac{839419}{90304}a^{19}+\frac{60237}{45152}a^{18}-\frac{15026675}{90304}a^{17}-\frac{1077413}{45152}a^{16}+\frac{9572773}{11288}a^{15}+\frac{1369849}{11288}a^{14}+\frac{654427}{1328}a^{13}+\frac{48163}{664}a^{12}-\frac{705450443}{45152}a^{11}-\frac{50596633}{22576}a^{10}+\frac{1944956765}{45152}a^{9}+\frac{138808993}{22576}a^{8}-\frac{460400139}{11288}a^{7}-\frac{65050465}{11288}a^{6}+\frac{208329553}{22576}a^{5}+\frac{14037093}{11288}a^{4}+\frac{34369179}{90304}a^{3}+\frac{3042773}{45152}a^{2}-\frac{8480327}{90304}a-\frac{623629}{45152}$, $\frac{186307}{10624}a^{19}+\frac{997587}{180608}a^{18}-\frac{3334899}{10624}a^{17}-\frac{17854115}{180608}a^{16}+\frac{4248267}{2656}a^{15}+\frac{22735759}{45152}a^{14}+\frac{618591}{664}a^{13}+\frac{97825}{332}a^{12}-\frac{156564033}{5312}a^{11}-\frac{838200689}{90304}a^{10}+\frac{431478959}{5312}a^{9}+\frac{2308212799}{90304}a^{8}-\frac{204119387}{2656}a^{7}-\frac{1090416095}{45152}a^{6}+\frac{11516737}{664}a^{5}+\frac{30589151}{5644}a^{4}+\frac{7468303}{10624}a^{3}+\frac{41433631}{180608}a^{2}-\frac{1850939}{10624}a-\frac{9824619}{180608}$, $\frac{511017}{90304}a^{19}-\frac{90205}{90304}a^{18}-\frac{9144961}{90304}a^{17}+\frac{1615121}{90304}a^{16}+\frac{2910633}{5644}a^{15}-\frac{2058907}{22576}a^{14}+\frac{402095}{1328}a^{13}-\frac{8733}{166}a^{12}-\frac{429349233}{45152}a^{11}+\frac{75811887}{45152}a^{10}+\frac{1181622639}{45152}a^{9}-\frac{209273161}{45152}a^{8}-\frac{139381451}{5644}a^{7}+\frac{99378863}{22576}a^{6}+\frac{124515001}{22576}a^{5}-\frac{5739171}{5644}a^{4}+\frac{21216649}{90304}a^{3}-\frac{2602017}{90304}a^{2}-\frac{5191213}{90304}a+\frac{868641}{90304}$, $\frac{613203}{90304}a^{19}+\frac{276005}{90304}a^{18}-\frac{10976461}{90304}a^{17}-\frac{4940779}{90304}a^{16}+\frac{3495811}{5644}a^{15}+\frac{3147419}{11288}a^{14}+\frac{478753}{1328}a^{13}+\frac{215281}{1328}a^{12}-\frac{515291635}{45152}a^{11}-\frac{231953829}{45152}a^{10}+\frac{1420303223}{45152}a^{9}+\frac{639441153}{45152}a^{8}-\frac{42011561}{1411}a^{7}-\frac{151339085}{11288}a^{6}+\frac{152084835}{22576}a^{5}+\frac{68485683}{22576}a^{4}+\frac{24274899}{90304}a^{3}+\frac{11061669}{90304}a^{2}-\frac{6433281}{90304}a-\frac{2864887}{90304}$, $\frac{1028941}{180608}a^{19}+\frac{353453}{180608}a^{18}-\frac{18418213}{180608}a^{17}-\frac{6326741}{180608}a^{16}+\frac{23463535}{45152}a^{15}+\frac{8059553}{45152}a^{14}+\frac{401599}{1328}a^{13}+\frac{68989}{664}a^{12}-\frac{864603075}{90304}a^{11}-\frac{296981543}{90304}a^{10}+\frac{2383173957}{90304}a^{9}+\frac{818584185}{90304}a^{8}-\frac{1128225199}{45152}a^{7}-\frac{387586297}{45152}a^{6}+\frac{128210353}{22576}a^{5}+\frac{22053889}{11288}a^{4}+\frac{35634857}{180608}a^{3}+\frac{11868017}{180608}a^{2}-\frac{9936021}{180608}a-\frac{3341997}{180608}$, $\frac{3152901}{180608}a^{19}+\frac{151697}{180608}a^{18}-\frac{56423513}{180608}a^{17}-\frac{2715261}{180608}a^{16}+\frac{71834125}{45152}a^{15}+\frac{3458537}{45152}a^{14}+\frac{620221}{664}a^{13}+\frac{29671}{664}a^{12}-\frac{2649146527}{90304}a^{11}-\frac{127481003}{90304}a^{10}+\frac{7290771749}{90304}a^{9}+\frac{351235553}{90304}a^{8}-\frac{3439297093}{45152}a^{7}-\frac{166028017}{45152}a^{6}+\frac{191451041}{11288}a^{5}+\frac{9303243}{11288}a^{4}+\frac{138430409}{180608}a^{3}+\frac{6845333}{180608}a^{2}-\frac{31174401}{180608}a-\frac{1735701}{180608}$, $\frac{7091077}{180608}a^{19}-\frac{21045}{180608}a^{18}-\frac{126931029}{180608}a^{17}+\frac{373597}{180608}a^{16}+\frac{161697441}{45152}a^{15}-\frac{466071}{45152}a^{14}+\frac{1384669}{664}a^{13}-\frac{10251}{1328}a^{12}-\frac{5958991279}{90304}a^{11}+\frac{17583651}{90304}a^{10}+\frac{16423261433}{90304}a^{9}-\frac{46135541}{90304}a^{8}-\frac{7770273393}{45152}a^{7}+\frac{19631263}{45152}a^{6}+\frac{438710267}{11288}a^{5}-\frac{1151673}{22576}a^{4}+\frac{283184841}{180608}a^{3}-\frac{2006817}{180608}a^{2}-\frac{70442637}{180608}a-\frac{33187}{180608}$ Copy content Toggle raw display (assuming GRH)
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:  \( 317973071.452 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{16}\cdot(2\pi)^{2}\cdot 317973071.452 \cdot 1}{2\cdot\sqrt{3056554299829282570240000000000}}\cr\approx \mathstrut & 0.235279428126 \end{aligned}\] (assuming GRH)

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 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1) 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 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1, 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 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1); 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 - 18*x^18 + 93*x^16 + 44*x^14 - 1686*x^12 + 4800*x^10 - 4846*x^8 + 1428*x^6 - 59*x^4 - 14*x^2 + 1); 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

$C_2^8.\POPlus(4,5)$ (as 20T1013):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
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 3686400
The 114 conjugacy class representatives for $C_2^8.\POPlus(4,5)$
Character table for $C_2^8.\POPlus(4,5)$

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.109268775200000.1

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 20 sibling: data not computed
Degree 32 sibling: data not computed
Degree 40 siblings: data not computed
Minimal sibling: 20.16.14753403803404679573577564160000000000.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.10.0.1}{10} }^{2}$ R ${\href{/padicField/7.10.0.1}{10} }^{2}$ ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ R ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.10.0.1}{10} }^{2}$ R R ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{4}$ ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.10.0.1}{10} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.4.0.1}{4} }^{2}$

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 2.4.1.0a1.1$x^{4} + x + 1$$1$$4$$0$$C_4$$$[\ ]^{4}$$
2.2.8.24b5.2$x^{16} + 8 x^{15} + 36 x^{14} + 112 x^{13} + 266 x^{12} + 506 x^{11} + 796 x^{10} + 1056 x^{9} + 1197 x^{8} + 1166 x^{7} + 978 x^{6} + 704 x^{5} + 432 x^{4} + 222 x^{3} + 92 x^{2} + 32 x + 7$$8$$2$$24$16T1825not computed
\(5\) Copy content Toggle raw display 5.2.2.2a1.2$x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$$2$$2$$2$$C_2^2$$$[\ ]_{2}^{2}$$
5.8.2.8a1.2$x^{16} + 2 x^{12} + 6 x^{10} + 8 x^{9} + 5 x^{8} + 6 x^{6} + 8 x^{5} + 13 x^{4} + 24 x^{3} + 28 x^{2} + 16 x + 9$$2$$8$$8$$C_8\times C_2$$$[\ ]_{2}^{8}$$
\(13\) Copy content Toggle raw display 13.6.1.0a1.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
13.6.1.0a1.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$$[\ ]^{6}$$
13.4.2.4a1.2$x^{8} + 6 x^{6} + 24 x^{5} + 13 x^{4} + 72 x^{3} + 156 x^{2} + 48 x + 17$$2$$4$$4$$C_4\times C_2$$$[\ ]_{2}^{4}$$
\(29\) Copy content Toggle raw display 29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.2.1.0a1.1$x^{2} + 24 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
29.1.3.2a1.1$x^{3} + 29$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
29.1.3.2a1.1$x^{3} + 29$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
29.5.1.0a1.1$x^{5} + 3 x + 27$$1$$5$$0$$C_5$$$[\ ]^{5}$$
29.5.1.0a1.1$x^{5} + 3 x + 27$$1$$5$$0$$C_5$$$[\ ]^{5}$$
\(31\) Copy content Toggle raw display 31.2.1.0a1.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
31.2.1.0a1.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
31.2.1.0a1.1$x^{2} + 29 x + 3$$1$$2$$0$$C_2$$$[\ ]^{2}$$
31.1.3.2a1.2$x^{3} + 93$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.1.3.2a1.2$x^{3} + 93$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.8.1.0a1.1$x^{8} + 25 x^{3} + 12 x^{2} + 24 x + 3$$1$$8$$0$$C_8$$$[\ ]^{8}$$

Spectrum of ring of integers

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