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Properties

Label 20.20.120...125.1
Degree $20$
Signature $(20, 0)$
Discriminant $1.206\times 10^{38}$
Root discriminant \(80.18\)
Ramified primes $5,23$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{20}$ (as 20T1)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^20 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399)
 
Copy content gp:K = bnfinit(y^20 - 120*y^18 + 6120*y^16 - 151*y^15 - 172800*y^14 + 13590*y^13 + 2948400*y^12 - 489240*y^11 - 31120079*y^10 + 8969400*y^9 + 199252740*y^8 - 88063200*y^7 - 720099540*y^6 + 442743929*y^5 + 1223413200*y^4 - 953469870*y^3 - 520959600*y^2 + 437027220*y - 64286399, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^20 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^20 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399)
 

\( x^{20} - 120 x^{18} + 6120 x^{16} - 151 x^{15} - 172800 x^{14} + 13590 x^{13} + 2948400 x^{12} + \cdots - 64286399 \) 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:  $(20, 0)$
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:   \(120567015877601807005703449249267578125\) \(\medspace = 5^{35}\cdot 23^{10}\) 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:  \(80.18\)
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:  $5^{7/4}23^{1/2}\approx 80.17914588789216$
Ramified primes:   \(5\), \(23\) 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:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_{20}$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(575=5^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{575}(1,·)$, $\chi_{575}(68,·)$, $\chi_{575}(137,·)$, $\chi_{575}(139,·)$, $\chi_{575}(461,·)$, $\chi_{575}(528,·)$, $\chi_{575}(22,·)$, $\chi_{575}(24,·)$, $\chi_{575}(346,·)$, $\chi_{575}(413,·)$, $\chi_{575}(482,·)$, $\chi_{575}(484,·)$, $\chi_{575}(231,·)$, $\chi_{575}(298,·)$, $\chi_{575}(367,·)$, $\chi_{575}(369,·)$, $\chi_{575}(116,·)$, $\chi_{575}(183,·)$, $\chi_{575}(252,·)$, $\chi_{575}(254,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{19}a^{10}-\frac{3}{19}a^{8}+\frac{6}{19}a^{6}-\frac{9}{19}a^{5}-\frac{8}{19}a^{4}+\frac{4}{19}a^{3}+\frac{5}{19}a^{2}-\frac{5}{19}a-\frac{5}{19}$, $\frac{1}{19}a^{11}-\frac{3}{19}a^{9}+\frac{6}{19}a^{7}-\frac{9}{19}a^{6}-\frac{8}{19}a^{5}+\frac{4}{19}a^{4}+\frac{5}{19}a^{3}-\frac{5}{19}a^{2}-\frac{5}{19}a$, $\frac{1}{19}a^{12}-\frac{3}{19}a^{8}-\frac{9}{19}a^{7}-\frac{9}{19}a^{6}-\frac{4}{19}a^{5}+\frac{7}{19}a^{3}-\frac{9}{19}a^{2}+\frac{4}{19}a+\frac{4}{19}$, $\frac{1}{920550931}a^{13}-\frac{9403020}{920550931}a^{12}-\frac{78}{920550931}a^{11}-\frac{1283246}{920550931}a^{10}+\frac{2340}{920550931}a^{9}+\frac{83097691}{920550931}a^{8}+\frac{15298242}{48450049}a^{7}+\frac{295580079}{920550931}a^{6}+\frac{339386215}{920550931}a^{5}+\frac{390432882}{920550931}a^{4}-\frac{388308008}{920550931}a^{3}+\frac{340244942}{920550931}a^{2}-\frac{386993864}{920550931}a-\frac{420265291}{920550931}$, $\frac{1}{920550931}a^{14}-\frac{84}{920550931}a^{12}-\frac{7968071}{920550931}a^{11}+\frac{2772}{920550931}a^{10}-\frac{18221484}{48450049}a^{9}-\frac{45360}{920550931}a^{8}+\frac{411638717}{920550931}a^{7}+\frac{387981416}{920550931}a^{6}+\frac{110972955}{920550931}a^{5}+\frac{337626247}{920550931}a^{4}-\frac{32626502}{920550931}a^{3}-\frac{191514052}{920550931}a^{2}+\frac{446132214}{920550931}a-\frac{559872}{920550931}$, $\frac{1}{920550931}a^{15}-\frac{22620967}{920550931}a^{12}-\frac{3780}{920550931}a^{11}-\frac{17950419}{920550931}a^{10}+\frac{151200}{920550931}a^{9}+\frac{75887362}{920550931}a^{8}+\frac{336700903}{920550931}a^{7}+\frac{327074699}{920550931}a^{6}-\frac{272411142}{920550931}a^{5}-\frac{182297734}{920550931}a^{4}+\frac{136646596}{920550931}a^{3}+\frac{295828285}{920550931}a^{2}-\frac{288761863}{920550931}a-\frac{321349066}{920550931}$, $\frac{1}{920550931}a^{16}-\frac{4320}{920550931}a^{12}+\frac{10265968}{920550931}a^{11}+\frac{190080}{920550931}a^{10}-\frac{237653709}{920550931}a^{9}-\frac{3499200}{920550931}a^{8}-\frac{128908906}{920550931}a^{7}-\frac{113997315}{920550931}a^{6}+\frac{5684463}{920550931}a^{5}-\frac{372887045}{920550931}a^{4}+\frac{313282282}{920550931}a^{3}-\frac{186046472}{920550931}a^{2}+\frac{328148180}{920550931}a-\frac{50388480}{920550931}$, $\frac{1}{920550931}a^{17}-\frac{9639370}{920550931}a^{12}-\frac{146880}{920550931}a^{11}-\frac{15720598}{920550931}a^{10}+\frac{6609600}{920550931}a^{9}-\frac{258646974}{920550931}a^{8}-\frac{17313790}{920550931}a^{7}-\frac{231665897}{920550931}a^{6}+\frac{452279799}{920550931}a^{5}+\frac{437126832}{920550931}a^{4}+\frac{293905985}{920550931}a^{3}+\frac{405611156}{920550931}a^{2}-\frac{46490166}{920550931}a-\frac{122721090}{920550931}$, $\frac{1}{920550931}a^{18}-\frac{176256}{920550931}a^{12}+\frac{7609326}{920550931}a^{11}+\frac{8724672}{920550931}a^{10}-\frac{280244008}{920550931}a^{9}-\frac{171320832}{920550931}a^{8}+\frac{447551765}{920550931}a^{7}-\frac{242107430}{920550931}a^{6}-\frac{441392882}{920550931}a^{5}-\frac{408976763}{920550931}a^{4}+\frac{182617407}{920550931}a^{3}-\frac{253398756}{920550931}a^{2}-\frac{42326560}{920550931}a+\frac{20519481}{920550931}$, $\frac{1}{920550931}a^{19}-\frac{257651}{920550931}a^{12}-\frac{264384}{48450049}a^{11}-\frac{4521958}{920550931}a^{10}+\frac{12690432}{48450049}a^{9}-\frac{68586578}{920550931}a^{8}-\frac{221873579}{920550931}a^{7}+\frac{291472965}{920550931}a^{6}-\frac{308363475}{920550931}a^{5}-\frac{384730637}{920550931}a^{4}-\frac{216088277}{920550931}a^{3}-\frac{331480628}{920550931}a^{2}-\frac{392139886}{920550931}a+\frac{128684722}{920550931}$ 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:  $19$
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:   $\frac{151}{48450049}a^{15}-\frac{13590}{48450049}a^{13}+\frac{489240}{48450049}a^{11}-\frac{15025}{48450049}a^{10}-\frac{8969400}{48450049}a^{9}+\frac{901500}{48450049}a^{8}+\frac{88063200}{48450049}a^{7}-\frac{18931500}{48450049}a^{6}-\frac{443838528}{48450049}a^{5}+\frac{162270000}{48450049}a^{4}+\frac{986307840}{48450049}a^{3}-\frac{486810000}{48450049}a^{2}-\frac{634055040}{48450049}a+\frac{185218751}{48450049}$, $\frac{12155}{920550931}a^{13}-\frac{42509}{920550931}a^{12}-\frac{948090}{920550931}a^{11}+\frac{3060648}{920550931}a^{10}+\frac{28442700}{920550931}a^{9}-\frac{82637496}{920550931}a^{8}-\frac{409574880}{920550931}a^{7}+\frac{1028377728}{920550931}a^{6}+\frac{2867024160}{920550931}a^{5}-\frac{5784624720}{920550931}a^{4}-\frac{8601072480}{920550931}a^{3}+\frac{11899799424}{920550931}a^{2}+\frac{7372347840}{920550931}a-\frac{3966599808}{920550931}$, $\frac{151}{920550931}a^{19}-\frac{906}{48450049}a^{17}-\frac{1296}{920550931}a^{16}+\frac{43488}{48450049}a^{15}+\frac{109391}{920550931}a^{14}-\frac{1141560}{48450049}a^{13}-\frac{3590124}{920550931}a^{12}+\frac{338554080}{920550931}a^{11}+\frac{56888172}{920550931}a^{10}-\frac{3202057529}{920550931}a^{9}-\frac{427012560}{920550931}a^{8}+\frac{17919874566}{920550931}a^{7}+\frac{1027853712}{920550931}a^{6}-\frac{55197866988}{920550931}a^{5}+\frac{3235463615}{920550931}a^{4}+\frac{79138483920}{920550931}a^{3}-\frac{16276558440}{920550931}a^{2}-\frac{1702716295}{48450049}a+\frac{472746600}{48450049}$, $\frac{85}{920550931}a^{19}-\frac{510}{48450049}a^{17}+\frac{24480}{48450049}a^{15}-\frac{642600}{48450049}a^{13}+\frac{10024560}{48450049}a^{11}-\frac{94517280}{48450049}a^{9}+\frac{523480320}{48450049}a^{7}-\frac{7431811}{920550931}a^{6}-\frac{1570440960}{48450049}a^{5}+\frac{267545196}{920550931}a^{4}+\frac{2141510400}{48450049}a^{3}-\frac{2407906764}{920550931}a^{2}-\frac{856604160}{48450049}a+\frac{3210542352}{920550931}$, $\frac{180}{920550931}a^{18}-\frac{755}{920550931}a^{17}-\frac{19440}{920550931}a^{16}+\frac{77010}{920550931}a^{15}+\frac{874800}{920550931}a^{14}-\frac{3261600}{920550931}a^{13}-\frac{21153355}{920550931}a^{12}+\frac{74201400}{920550931}a^{11}+\frac{294822360}{920550931}a^{10}-\frac{978480000}{920550931}a^{9}-\frac{2345482260}{920550931}a^{8}+\frac{7497511645}{920550931}a^{7}+\frac{510131520}{48450049}a^{6}-\frac{31331985090}{920550931}a^{5}-\frac{15039529200}{920550931}a^{4}+\frac{59541906071}{920550931}a^{3}-\frac{5645892475}{920550931}a^{2}-\frac{28896033078}{920550931}a+\frac{393955500}{48450049}$, $\frac{151}{920550931}a^{18}-\frac{216}{920550931}a^{17}-\frac{16308}{920550931}a^{16}+\frac{22032}{920550931}a^{15}+\frac{733860}{920550931}a^{14}-\frac{940369}{920550931}a^{13}-\frac{17775720}{920550931}a^{12}+\frac{21793902}{920550931}a^{11}+\frac{249512400}{920550931}a^{10}-\frac{296898660}{920550931}a^{9}-\frac{2027881529}{920550931}a^{8}+\frac{2387566152}{920550931}a^{7}+\frac{8922860592}{920550931}a^{6}-\frac{10603144944}{920550931}a^{5}-\frac{17599484880}{920550931}a^{4}+\frac{21282898655}{920550931}a^{3}+\frac{7872306821}{920550931}a^{2}-\frac{9501487470}{920550931}a-\frac{69802332}{920550931}$, $\frac{29}{920550931}a^{18}-\frac{3132}{920550931}a^{16}+\frac{140940}{920550931}a^{14}-\frac{3420144}{920550931}a^{12}+\frac{48370608}{920550931}a^{10}-\frac{401848128}{920550931}a^{8}-\frac{2227595}{920550931}a^{7}+\frac{1875291264}{920550931}a^{6}+\frac{93558990}{920550931}a^{5}-\frac{4383797760}{920550931}a^{4}-\frac{1122707880}{920550931}a^{3}+\frac{3945417984}{920550931}a^{2}+\frac{3368123640}{920550931}a-\frac{584506368}{920550931}$, $\frac{66}{920550931}a^{19}-\frac{396}{48450049}a^{17}-\frac{1661}{920550931}a^{16}+\frac{19008}{48450049}a^{15}+\frac{149490}{920550931}a^{14}-\frac{498960}{48450049}a^{13}-\frac{5381640}{920550931}a^{12}+\frac{148057019}{920550931}a^{11}+\frac{98663400}{920550931}a^{10}-\frac{1404324372}{920550931}a^{9}-\frac{968695200}{920550931}a^{8}+\frac{7931120868}{920550931}a^{7}+\frac{4870183219}{920550931}a^{6}-\frac{24953593104}{920550931}a^{5}-\frac{10464375085}{920550931}a^{4}+\frac{36948486960}{920550931}a^{3}+\frac{4236255780}{920550931}a^{2}-\frac{14674837045}{920550931}a+\frac{1713130920}{920550931}$, $\frac{5059}{920550931}a^{14}-\frac{424956}{920550931}a^{12}-\frac{30421}{920550931}a^{11}+\frac{14023548}{920550931}a^{10}+\frac{2007786}{920550931}a^{9}-\frac{229476240}{920550931}a^{8}-\frac{48186864}{920550931}a^{7}+\frac{1927600416}{920550931}a^{6}+\frac{505962072}{920550931}a^{5}-\frac{7710401664}{920550931}a^{4}-\frac{2168408880}{920550931}a^{3}+\frac{11565602496}{920550931}a^{2}+\frac{2602090656}{920550931}a-\frac{2832392448}{920550931}$, $\frac{255}{920550931}a^{19}+\frac{396}{920550931}a^{18}-\frac{29827}{920550931}a^{17}-\frac{43133}{920550931}a^{16}+\frac{1462608}{920550931}a^{15}+\frac{1937327}{920550931}a^{14}-\frac{39034044}{920550931}a^{13}-\frac{46033009}{920550931}a^{12}+\frac{616362368}{920550931}a^{11}+\frac{615611766}{920550931}a^{10}-\frac{5873681605}{920550931}a^{9}-\frac{4522455299}{920550931}a^{8}+\frac{33139567681}{920550931}a^{7}+\frac{16019056767}{920550931}a^{6}-\frac{104440262683}{920550931}a^{5}-\frac{15484239324}{920550931}a^{4}+\frac{160790859024}{920550931}a^{3}-\frac{28466136697}{920550931}a^{2}-\frac{4124528496}{48450049}a+\frac{26228841946}{920550931}$, $\frac{189}{920550931}a^{19}-\frac{719}{920550931}a^{18}-\frac{20617}{920550931}a^{17}+\frac{80043}{920550931}a^{16}+\frac{926615}{920550931}a^{15}-\frac{3724028}{920550931}a^{14}-\frac{21943285}{920550931}a^{13}+\frac{4926979}{48450049}a^{12}+\frac{287689113}{920550931}a^{11}-\frac{1369534259}{920550931}a^{10}-\frac{1935773962}{920550931}a^{9}+\frac{11645744173}{920550931}a^{8}+\frac{4068812177}{920550931}a^{7}-\frac{53628899728}{920550931}a^{6}+\frac{19053819629}{920550931}a^{5}+\frac{109328166241}{920550931}a^{4}-\frac{88952682617}{920550931}a^{3}-\frac{45861102653}{920550931}a^{2}+\frac{54182247482}{920550931}a-\frac{12590845779}{920550931}$, $\frac{397}{920550931}a^{19}-\frac{45084}{920550931}a^{17}-\frac{6392}{920550931}a^{16}+\frac{2167471}{920550931}a^{15}+\frac{561461}{920550931}a^{14}-\frac{57380150}{920550931}a^{13}-\frac{19664723}{920550931}a^{12}+\frac{911112449}{920550931}a^{11}+\frac{348413015}{920550931}a^{10}-\frac{8859687614}{920550931}a^{9}-\frac{3250998071}{920550931}a^{8}+\frac{51589968264}{920550931}a^{7}+\frac{14748910309}{920550931}a^{6}-\frac{166894277735}{920550931}a^{5}-\frac{22038611799}{920550931}a^{4}+\frac{248619178770}{920550931}a^{3}-\frac{23478017220}{920550931}a^{2}-\frac{90496456952}{920550931}a+\frac{27951302223}{920550931}$, $\frac{2870}{920550931}a^{15}-\frac{258300}{920550931}a^{13}+\frac{9298800}{920550931}a^{11}+\frac{356148}{920550931}a^{10}-\frac{170478000}{920550931}a^{9}-\frac{21368880}{920550931}a^{8}+\frac{1673784000}{920550931}a^{7}+\frac{448746480}{920550931}a^{6}-\frac{8387421311}{920550931}a^{5}-\frac{3846398400}{920550931}a^{4}+\frac{17292879330}{920550931}a^{3}+\frac{11539195200}{920550931}a^{2}-\frac{3330235980}{920550931}a-\frac{1372109482}{920550931}$, $\frac{397}{920550931}a^{19}+\frac{568}{920550931}a^{18}-\frac{2382}{48450049}a^{17}-\frac{65232}{920550931}a^{16}+\frac{2159549}{920550931}a^{15}+\frac{3083594}{920550931}a^{14}-\frac{55912439}{920550931}a^{13}-\frac{77260134}{920550931}a^{12}+\frac{851939059}{920550931}a^{11}+\frac{1100063869}{920550931}a^{10}-\frac{7762118481}{920550931}a^{9}-\frac{8857093596}{920550931}a^{8}+\frac{41190192937}{920550931}a^{7}+\frac{37724724589}{920550931}a^{6}-\frac{118290941347}{920550931}a^{5}-\frac{72100197687}{920550931}a^{4}+\frac{8130711810}{48450049}a^{3}+\frac{35921074716}{920550931}a^{2}-\frac{51570334587}{920550931}a+\frac{8273516481}{920550931}$, $\frac{256}{920550931}a^{19}-\frac{143}{920550931}a^{18}-\frac{30306}{920550931}a^{17}+\frac{20792}{920550931}a^{16}+\frac{1491796}{920550931}a^{15}-\frac{1220999}{920550931}a^{14}-\frac{39458265}{920550931}a^{13}+\frac{37954411}{920550931}a^{12}+\frac{604079049}{920550931}a^{11}-\frac{677964309}{920550931}a^{10}-\frac{5356089083}{920550931}a^{9}+\frac{6987178963}{920550931}a^{8}+\frac{25874737255}{920550931}a^{7}-\frac{39049950662}{920550931}a^{6}-\frac{58390463360}{920550931}a^{5}+\frac{100762466619}{920550931}a^{4}+\frac{44156633454}{920550931}a^{3}-\frac{82678667701}{920550931}a^{2}-\frac{16006143266}{920550931}a+\frac{15203400408}{920550931}$, $\frac{6}{48450049}a^{19}-\frac{7}{920550931}a^{18}-\frac{12845}{920550931}a^{17}-\frac{92}{48450049}a^{16}+\frac{613465}{920550931}a^{15}+\frac{206364}{920550931}a^{14}-\frac{16141485}{920550931}a^{13}-\frac{8691976}{920550931}a^{12}+\frac{254155986}{920550931}a^{11}+\frac{188789682}{920550931}a^{10}-\frac{2426187389}{920550931}a^{9}-\frac{2311534044}{920550931}a^{8}+\frac{13461411362}{920550931}a^{7}+\frac{15855726419}{920550931}a^{6}-\frac{38162606052}{920550931}a^{5}-\frac{55263525228}{920550931}a^{4}+\frac{37198728532}{920550931}a^{3}+\frac{72983224523}{920550931}a^{2}+\frac{6058637064}{920550931}a-\frac{14251944400}{920550931}$, $\frac{302}{920550931}a^{19}-\frac{64}{920550931}a^{18}-\frac{35116}{920550931}a^{17}+\frac{1086}{920550931}a^{16}+\frac{1724910}{920550931}a^{15}+\frac{218206}{920550931}a^{14}-\frac{46508747}{920550931}a^{13}-\frac{11773056}{920550931}a^{12}+\frac{39406134}{48450049}a^{11}+\frac{255366205}{920550931}a^{10}-\frac{7331123537}{920550931}a^{9}-\frac{2803106948}{920550931}a^{8}+\frac{42522877029}{920550931}a^{7}+\frac{15739986876}{920550931}a^{6}-\frac{134498611746}{920550931}a^{5}-\frac{40175970674}{920550931}a^{4}+\frac{189123194041}{920550931}a^{3}+\frac{29257343995}{920550931}a^{2}-\frac{60407160913}{920550931}a+\frac{10069839160}{920550931}$, $\frac{265}{920550931}a^{19}+\frac{1359}{920550931}a^{18}-\frac{32154}{920550931}a^{17}-\frac{154901}{920550931}a^{16}+\frac{1659844}{920550931}a^{15}+\frac{7370099}{920550931}a^{14}-\frac{47554717}{920550931}a^{13}-\frac{189099759}{920550931}a^{12}+\frac{826859382}{920550931}a^{11}+\frac{2816953544}{920550931}a^{10}-\frac{8947082744}{920550931}a^{9}-\frac{24347970633}{920550931}a^{8}+\frac{59068314720}{920550931}a^{7}+\frac{114258847211}{920550931}a^{6}-\frac{219921238944}{920550931}a^{5}-\frac{241766065357}{920550931}a^{4}+\frac{375740652951}{920550931}a^{3}+\frac{117053446814}{920550931}a^{2}-\frac{147860640787}{920550931}a+\frac{24602592260}{920550931}$, $\frac{690}{920550931}a^{19}-\frac{1985}{920550931}a^{18}-\frac{77081}{920550931}a^{17}+\frac{221012}{920550931}a^{16}+\frac{3590311}{920550931}a^{15}-\frac{10329151}{920550931}a^{14}-\frac{89902466}{920550931}a^{13}+\frac{262178073}{920550931}a^{12}+\frac{1296978030}{920550931}a^{11}-\frac{3897681475}{920550931}a^{10}-\frac{10646437379}{920550931}a^{9}+\frac{33979226031}{920550931}a^{8}+\frac{44871854481}{920550931}a^{7}-\frac{163118018664}{920550931}a^{6}-\frac{67526539819}{920550931}a^{5}+\frac{365119148063}{920550931}a^{4}-\frac{37411936449}{920550931}a^{3}-\frac{12870533750}{48450049}a^{2}+\frac{34424801472}{920550931}a+\frac{19230823717}{920550931}$ 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:  \( 3975952893490 \) (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)
 
Unit signature rank:  \( 19 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{20}\cdot(2\pi)^{0}\cdot 3975952893490 \cdot 1}{2\cdot\sqrt{120567015877601807005703449249267578125}}\cr\approx \mathstrut & 0.189844007031237 \end{aligned}\] (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 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399) 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 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399, 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 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399); 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 - 120*x^18 + 6120*x^16 - 151*x^15 - 172800*x^14 + 13590*x^13 + 2948400*x^12 - 489240*x^11 - 31120079*x^10 + 8969400*x^9 + 199252740*x^8 - 88063200*x^7 - 720099540*x^6 + 442743929*x^5 + 1223413200*x^4 - 953469870*x^3 - 520959600*x^2 + 437027220*x - 64286399); 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

$C_{20}$ (as 20T1):

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 cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{230 -46 \sqrt{5}})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\)

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20$ $20$ R ${\href{/padicField/7.4.0.1}{4} }^{5}$ ${\href{/padicField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/padicField/19.5.0.1}{5} }^{4}$ R ${\href{/padicField/29.10.0.1}{10} }^{2}$ ${\href{/padicField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/padicField/41.5.0.1}{5} }^{4}$ ${\href{/padicField/43.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/padicField/59.10.0.1}{10} }^{2}$

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
\(5\) Copy content Toggle raw display 5.1.20.35a1.500$x^{20} + 5 x^{16} + 120$$20$$1$$35$20T1not computed
\(23\) Copy content Toggle raw display 23.10.2.10a1.1$x^{20} + 34 x^{15} + 10 x^{14} + 30 x^{13} + 12 x^{12} + 2 x^{11} + 299 x^{10} + 170 x^{9} + 535 x^{8} + 354 x^{7} + 319 x^{6} + 360 x^{5} + 116 x^{4} + 162 x^{3} + 61 x^{2} + 33 x + 25$$2$$10$$10$20T1$$[\ ]_{2}^{10}$$

Spectrum of ring of integers

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