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Properties

Label 20.0.25891569508...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 23^{10}$
Root discriminant $209.25$
Ramified primes $2, 5, 23$
Class number $191765500$ (GRH)
Class group $[10, 19176550]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![94712038799, -31427128530, 144931284995, -6982300200, 44291166105, 5240050776, 3728327265, 888713020, 258898250, 161576710, 32510275, -8013640, 6805195, -657950, 475640, -11102, 14850, -60, 200, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 200*x^18 - 60*x^17 + 14850*x^16 - 11102*x^15 + 475640*x^14 - 657950*x^13 + 6805195*x^12 - 8013640*x^11 + 32510275*x^10 + 161576710*x^9 + 258898250*x^8 + 888713020*x^7 + 3728327265*x^6 + 5240050776*x^5 + 44291166105*x^4 - 6982300200*x^3 + 144931284995*x^2 - 31427128530*x + 94712038799)
 
gp: K = bnfinit(x^20 + 200*x^18 - 60*x^17 + 14850*x^16 - 11102*x^15 + 475640*x^14 - 657950*x^13 + 6805195*x^12 - 8013640*x^11 + 32510275*x^10 + 161576710*x^9 + 258898250*x^8 + 888713020*x^7 + 3728327265*x^6 + 5240050776*x^5 + 44291166105*x^4 - 6982300200*x^3 + 144931284995*x^2 - 31427128530*x + 94712038799, 1)
 

Normalized defining polynomial

\( x^{20} + 200 x^{18} - 60 x^{17} + 14850 x^{16} - 11102 x^{15} + 475640 x^{14} - 657950 x^{13} + 6805195 x^{12} - 8013640 x^{11} + 32510275 x^{10} + 161576710 x^{9} + 258898250 x^{8} + 888713020 x^{7} + 3728327265 x^{6} + 5240050776 x^{5} + 44291166105 x^{4} - 6982300200 x^{3} + 144931284995 x^{2} - 31427128530 x + 94712038799 \)

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:  \(25891569508530625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $209.25$
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:  $2, 5, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4600=2^{3}\cdot 5^{2}\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{4600}(1,·)$, $\chi_{4600}(3909,·)$, $\chi_{4600}(3681,·)$, $\chi_{4600}(1609,·)$, $\chi_{4600}(461,·)$, $\chi_{4600}(4141,·)$, $\chi_{4600}(4369,·)$, $\chi_{4600}(3221,·)$, $\chi_{4600}(921,·)$, $\chi_{4600}(1381,·)$, $\chi_{4600}(2529,·)$, $\chi_{4600}(229,·)$, $\chi_{4600}(689,·)$, $\chi_{4600}(2989,·)$, $\chi_{4600}(2301,·)$, $\chi_{4600}(1841,·)$, $\chi_{4600}(2761,·)$, $\chi_{4600}(3449,·)$, $\chi_{4600}(1149,·)$, $\chi_{4600}(2069,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{49} a^{13} - \frac{1}{49} a^{12} + \frac{1}{49} a^{11} - \frac{1}{49} a^{10} + \frac{1}{49} a^{9} - \frac{1}{49} a^{8} - \frac{1}{49} a^{7} + \frac{8}{49} a^{6} - \frac{8}{49} a^{5} + \frac{8}{49} a^{4} - \frac{8}{49} a^{3} + \frac{8}{49} a^{2} - \frac{1}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{49} a^{15} - \frac{2}{49} a^{9} + \frac{1}{49} a^{3}$, $\frac{1}{49} a^{16} - \frac{2}{49} a^{10} + \frac{1}{49} a^{4}$, $\frac{1}{49} a^{17} - \frac{2}{49} a^{11} + \frac{1}{49} a^{5}$, $\frac{1}{208396930943896116071699} a^{18} + \frac{1684450865135907223932}{208396930943896116071699} a^{17} + \frac{624092801396920044914}{208396930943896116071699} a^{16} - \frac{1299897929952836697964}{208396930943896116071699} a^{15} + \frac{1639070039607538164103}{208396930943896116071699} a^{14} + \frac{130662579470329292420}{208396930943896116071699} a^{13} - \frac{3505336828895777016233}{208396930943896116071699} a^{12} - \frac{1230156949211976905391}{29770990134842302295957} a^{11} + \frac{1611736928522687112304}{29770990134842302295957} a^{10} - \frac{10590372557597608767675}{208396930943896116071699} a^{9} - \frac{2507608754012423683406}{208396930943896116071699} a^{8} + \frac{4949245614670307790393}{208396930943896116071699} a^{7} - \frac{7709174783709841688199}{29770990134842302295957} a^{6} + \frac{52564652725011148812472}{208396930943896116071699} a^{5} + \frac{75468529333469748795266}{208396930943896116071699} a^{4} + \frac{6018887033443632985276}{208396930943896116071699} a^{3} + \frac{9649090346822623670846}{29770990134842302295957} a^{2} + \frac{307311485832919569955}{4252998590691757470851} a + \frac{211101161270792426320}{607571227241679638693}$, $\frac{1}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{19} - \frac{147916906431614235438830873560881625139775401930811472953}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{18} + \frac{292715920165102041468809020338088255055472871890502538789826458218643481210325}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{17} + \frac{930255256118517495441043595526394186110990633353709378172746542920701203141804}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{16} - \frac{178276586876037117617621242462553513326888530039032120590872091186149411944676}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{15} - \frac{529858116811278694593483438142995707223244515409904052246439625603120687602070}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{14} - \frac{327794526248160610265168145738416171509250011508793444211846924569932064901620}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{13} - \frac{589514549342506199657730565282047903337149490356942545965885279815140980194938}{14095380088925440884449027408090118744042700710830106422477967514252739001007351} a^{12} + \frac{225422259206290489739530967012572324714531933968310627052566205762166471351502}{14095380088925440884449027408090118744042700710830106422477967514252739001007351} a^{11} + \frac{3851459837515524449712760632747366095452935302920447356131655198281104387445869}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{10} + \frac{6000516267314893302566148832073250537207067704427784000175221085770487174458696}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{9} - \frac{3017320175681646766326627423842432877208166650077812253108298449277787372590609}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{8} - \frac{543813812433667273696339780953131487945976352140300741948889629326374135495252}{14095380088925440884449027408090118744042700710830106422477967514252739001007351} a^{7} + \frac{259560147485562772736288643513209717359191987821644483269231435489268397715052}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{6} + \frac{46961165737148636911499642398690584698391004755015628973826668888235797253833583}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{5} + \frac{27322831377430473934388945253830868293286176652534567239239346498979105789804904}{98667660622478086191143191856630831208298904975810744957345772599769173007051457} a^{4} + \frac{1340194361155908909459000465739375629649947118157493440320659028425500422680472}{14095380088925440884449027408090118744042700710830106422477967514252739001007351} a^{3} - \frac{14349042183972314589427139836060325904973040162373410753737190524961598767396}{287660818141335528254061783838573851919238790016940947397509541107198755122599} a^{2} - \frac{139482430914528811543808443246212660497560372118146509717689507286638868597203}{287660818141335528254061783838573851919238790016940947397509541107198755122599} a + \frac{18207089756001950332406037950018875260687742330183149294813642471987931842928}{41094402591619361179151683405510550274176970002420135342501363015314107874657}$

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

Class group and class number

$C_{10}\times C_{19176550}$, which has order $191765500$ (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:  \( 42294001.73672045 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-230}) \), \(\Q(\sqrt{-115}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{2}, \sqrt{-115})\), 5.5.390625.1, 10.0.160908575000000000000000.1, 10.0.4910540008544921875.3, 10.10.5000000000000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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

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
$23$23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
23.10.5.1$x^{10} - 1058 x^{6} + 279841 x^{2} - 25745372$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$