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Properties

Label 20.0.12930852715...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{10}\cdot 61^{16}$
Root discriminant $403.26$
Ramified primes $2, 5, 61$
Class number $1156287922$ (GRH)
Class group $[1156287922]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![429330790631, 29552104068, 238384484610, -51654328704, 53053915879, -14978990136, 9530828238, -1079824928, 510298027, -21392148, 2083316, 767904, 37504, -82176, 30386, -3948, 1159, -8, 10, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 10*x^18 - 8*x^17 + 1159*x^16 - 3948*x^15 + 30386*x^14 - 82176*x^13 + 37504*x^12 + 767904*x^11 + 2083316*x^10 - 21392148*x^9 + 510298027*x^8 - 1079824928*x^7 + 9530828238*x^6 - 14978990136*x^5 + 53053915879*x^4 - 51654328704*x^3 + 238384484610*x^2 + 29552104068*x + 429330790631)
 
gp: K = bnfinit(x^20 - 4*x^19 + 10*x^18 - 8*x^17 + 1159*x^16 - 3948*x^15 + 30386*x^14 - 82176*x^13 + 37504*x^12 + 767904*x^11 + 2083316*x^10 - 21392148*x^9 + 510298027*x^8 - 1079824928*x^7 + 9530828238*x^6 - 14978990136*x^5 + 53053915879*x^4 - 51654328704*x^3 + 238384484610*x^2 + 29552104068*x + 429330790631, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 10 x^{18} - 8 x^{17} + 1159 x^{16} - 3948 x^{15} + 30386 x^{14} - 82176 x^{13} + 37504 x^{12} + 767904 x^{11} + 2083316 x^{10} - 21392148 x^{9} + 510298027 x^{8} - 1079824928 x^{7} + 9530828238 x^{6} - 14978990136 x^{5} + 53053915879 x^{4} - 51654328704 x^{3} + 238384484610 x^{2} + 29552104068 x + 429330790631 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12930852715467736934189031479041360155115520000000000=2^{55}\cdot 5^{10}\cdot 61^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $403.26$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 5, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4880=2^{4}\cdot 5\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4880}(1,·)$, $\chi_{4880}(3779,·)$, $\chi_{4880}(2779,·)$, $\chi_{4880}(2441,·)$, $\chi_{4880}(1099,·)$, $\chi_{4880}(81,·)$, $\chi_{4880}(339,·)$, $\chi_{4880}(4121,·)$, $\chi_{4880}(2521,·)$, $\chi_{4880}(1179,·)$, $\chi_{4880}(3059,·)$, $\chi_{4880}(3619,·)$, $\chi_{4880}(1681,·)$, $\chi_{4880}(1961,·)$, $\chi_{4880}(619,·)$, $\chi_{4880}(4401,·)$, $\chi_{4880}(3539,·)$, $\chi_{4880}(241,·)$, $\chi_{4880}(2681,·)$, $\chi_{4880}(1339,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{5} a^{10} - \frac{2}{5} a^{9} - \frac{2}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} + \frac{1}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2} - \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{13} - \frac{2}{5} a^{9} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{2}{5} a^{6} - \frac{2}{5} a^{5} - \frac{1}{5} a^{4} + \frac{1}{5} a^{3} - \frac{1}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{1}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{299135} a^{16} + \frac{10653}{299135} a^{15} + \frac{23764}{299135} a^{14} + \frac{32}{2063} a^{13} - \frac{17837}{299135} a^{12} + \frac{23974}{299135} a^{11} + \frac{3268}{59827} a^{10} + \frac{57468}{299135} a^{9} - \frac{117076}{299135} a^{8} + \frac{127968}{299135} a^{7} + \frac{24148}{59827} a^{6} - \frac{52979}{299135} a^{5} + \frac{32832}{299135} a^{4} + \frac{60469}{299135} a^{3} + \frac{23021}{59827} a^{2} + \frac{106564}{299135} a - \frac{10598}{59827}$, $\frac{1}{14059345} a^{17} - \frac{19}{14059345} a^{16} + \frac{1202788}{14059345} a^{15} - \frac{8239}{96961} a^{14} - \frac{896566}{14059345} a^{13} - \frac{1065962}{14059345} a^{12} - \frac{911341}{14059345} a^{11} - \frac{346096}{14059345} a^{10} - \frac{983031}{2811869} a^{9} + \frac{2349571}{14059345} a^{8} - \frac{899886}{14059345} a^{7} - \frac{198713}{2811869} a^{6} - \frac{1017316}{14059345} a^{5} - \frac{215031}{14059345} a^{4} + \frac{531304}{2811869} a^{3} + \frac{1150854}{14059345} a^{2} - \frac{6872833}{14059345} a - \frac{330}{2063}$, $\frac{1}{550616356459272170769795538015} a^{18} + \frac{15675890406847527213549}{550616356459272170769795538015} a^{17} - \frac{756154484210009876036433}{550616356459272170769795538015} a^{16} + \frac{19027163098335320470913157867}{550616356459272170769795538015} a^{15} - \frac{9416592323781827231140278438}{550616356459272170769795538015} a^{14} + \frac{35668889397849612658311435408}{550616356459272170769795538015} a^{13} - \frac{7642119842904814852854857710}{110123271291854434153959107603} a^{12} - \frac{44274505133404289675106553941}{550616356459272170769795538015} a^{11} + \frac{5865496775079916560558333747}{550616356459272170769795538015} a^{10} - \frac{1803259261544028139824373253}{550616356459272170769795538015} a^{9} + \frac{248880846460092615634848369718}{550616356459272170769795538015} a^{8} + \frac{31678882997016690935419724775}{110123271291854434153959107603} a^{7} + \frac{221501624641190288997933210122}{550616356459272170769795538015} a^{6} + \frac{19514411672744648412813183582}{110123271291854434153959107603} a^{5} - \frac{95317189091188797151018409732}{550616356459272170769795538015} a^{4} - \frac{166418466304911444214224334161}{550616356459272170769795538015} a^{3} - \frac{19866986154364390319072268288}{550616356459272170769795538015} a^{2} + \frac{33784777123565286891931052796}{550616356459272170769795538015} a - \frac{3720256682012615925143712811}{11715241626793024909995649745}$, $\frac{1}{222812559804031963980676184637589901697888556424530749168112145} a^{19} - \frac{146578481296867897107101687236223}{222812559804031963980676184637589901697888556424530749168112145} a^{18} - \frac{5914663118413252619720252874941096286244823046453787302}{222812559804031963980676184637589901697888556424530749168112145} a^{17} + \frac{320834587616047228100579287732500605715919058624954270741}{222812559804031963980676184637589901697888556424530749168112145} a^{16} - \frac{21951634708259902712795868425076275935232637481849794710091777}{222812559804031963980676184637589901697888556424530749168112145} a^{15} + \frac{8734374979042976925607854843332032705728845360930286231231952}{222812559804031963980676184637589901697888556424530749168112145} a^{14} - \frac{2332755086865349363237036537364624642054254644934328765091917}{44562511960806392796135236927517980339577711284906149833622429} a^{13} + \frac{116349686353251076938921024229831880726215844707913806945964}{7683191717380412551057799470261720748203053669811405143728005} a^{12} + \frac{18397712528997851330375512321679408127774058978039060389700958}{222812559804031963980676184637589901697888556424530749168112145} a^{11} - \frac{644870494384799993699422871968746283285568663867240070444743}{7683191717380412551057799470261720748203053669811405143728005} a^{10} - \frac{69658725169748725691889736641429695496941838401926665459425193}{222812559804031963980676184637589901697888556424530749168112145} a^{9} + \frac{102291109357190008253457002985248174747123478086886861778290873}{222812559804031963980676184637589901697888556424530749168112145} a^{8} - \frac{28419547684563183340542453492595940702807583423989178875094}{948138552357582825449685892074850645522930027338428719864307} a^{7} - \frac{76668545317181095776333062318294322024599523019979125504276836}{222812559804031963980676184637589901697888556424530749168112145} a^{6} + \frac{90346859147014572021154942599464605904265982439992043672462033}{222812559804031963980676184637589901697888556424530749168112145} a^{5} + \frac{365278892265756893641585676063532675854602628942344786393519}{1536638343476082510211559894052344149640610733962281028745601} a^{4} + \frac{25207588106141979192461467003440315907912293185466533395723093}{222812559804031963980676184637589901697888556424530749168112145} a^{3} + \frac{76891593590117832608023313057587038917386671868407762901081344}{222812559804031963980676184637589901697888556424530749168112145} a^{2} + \frac{23547053638325763962612130217473193843870145455352248303912031}{222812559804031963980676184637589901697888556424530749168112145} a + \frac{369191854855232254912327819580933619461000251923012086827481}{948138552357582825449685892074850645522930027338428719864307}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{1156287922}$, which has order $1156287922$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 421902771.2761871 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.0.51200.2, 5.5.13845841.1, 10.10.6281865232294903808.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{20}$ $20$ $20$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
5Data not computed
61Data not computed