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Properties

Label 20.0.70103795866...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{15}\cdot 41^{16}$
Root discriminant $438.82$
Ramified primes $2, 5, 41$
Class number $24298398100$ (GRH)
Class group $[24298398100]$ (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![465058055151, 115448641956, 221685746346, 38060451720, 46311595263, 5435837856, 5662159086, 329523584, 433071955, -5463084, 24013092, -1901616, 1213272, -122536, 57274, -4884, 1959, -192, 42, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 42*x^18 - 192*x^17 + 1959*x^16 - 4884*x^15 + 57274*x^14 - 122536*x^13 + 1213272*x^12 - 1901616*x^11 + 24013092*x^10 - 5463084*x^9 + 433071955*x^8 + 329523584*x^7 + 5662159086*x^6 + 5435837856*x^5 + 46311595263*x^4 + 38060451720*x^3 + 221685746346*x^2 + 115448641956*x + 465058055151)
 
gp: K = bnfinit(x^20 - 4*x^19 + 42*x^18 - 192*x^17 + 1959*x^16 - 4884*x^15 + 57274*x^14 - 122536*x^13 + 1213272*x^12 - 1901616*x^11 + 24013092*x^10 - 5463084*x^9 + 433071955*x^8 + 329523584*x^7 + 5662159086*x^6 + 5435837856*x^5 + 46311595263*x^4 + 38060451720*x^3 + 221685746346*x^2 + 115448641956*x + 465058055151, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 42 x^{18} - 192 x^{17} + 1959 x^{16} - 4884 x^{15} + 57274 x^{14} - 122536 x^{13} + 1213272 x^{12} - 1901616 x^{11} + 24013092 x^{10} - 5463084 x^{9} + 433071955 x^{8} + 329523584 x^{7} + 5662159086 x^{6} + 5435837856 x^{5} + 46311595263 x^{4} + 38060451720 x^{3} + 221685746346 x^{2} + 115448641956 x + 465058055151 \)

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:  \(70103795866390485914736344924648636416000000000000000=2^{55}\cdot 5^{15}\cdot 41^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.82$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3280=2^{4}\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{3280}(1,·)$, $\chi_{3280}(133,·)$, $\chi_{3280}(961,·)$, $\chi_{3280}(1609,·)$, $\chi_{3280}(1281,·)$, $\chi_{3280}(1677,·)$, $\chi_{3280}(3117,·)$, $\chi_{3280}(1041,·)$, $\chi_{3280}(3093,·)$, $\chi_{3280}(1369,·)$, $\chi_{3280}(1117,·)$, $\chi_{3280}(329,·)$, $\chi_{3280}(3173,·)$, $\chi_{3280}(877,·)$, $\chi_{3280}(797,·)$, $\chi_{3280}(1841,·)$, $\chi_{3280}(693,·)$, $\chi_{3280}(1289,·)$, $\chi_{3280}(2169,·)$, $\chi_{3280}(2133,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} - \frac{1}{27} a^{8} + \frac{1}{27} a^{6} - \frac{10}{81} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{729} a^{14} - \frac{1}{729} a^{13} - \frac{1}{729} a^{12} + \frac{4}{729} a^{11} + \frac{1}{243} a^{10} + \frac{1}{243} a^{9} + \frac{32}{729} a^{8} + \frac{13}{729} a^{7} - \frac{20}{729} a^{6} - \frac{46}{729} a^{5} + \frac{31}{243} a^{4} - \frac{2}{27} a^{3} - \frac{4}{27} a^{2} + \frac{1}{9} a$, $\frac{1}{6561} a^{15} - \frac{1}{2187} a^{14} - \frac{2}{6561} a^{13} + \frac{1}{729} a^{12} + \frac{25}{6561} a^{11} - \frac{2}{2187} a^{10} + \frac{26}{6561} a^{9} + \frac{34}{2187} a^{8} - \frac{250}{6561} a^{7} - \frac{2}{729} a^{6} + \frac{812}{6561} a^{5} + \frac{71}{2187} a^{4} + \frac{13}{729} a^{3} + \frac{70}{243} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{6561} a^{16} - \frac{2}{6561} a^{14} - \frac{2}{2187} a^{13} - \frac{11}{6561} a^{12} + \frac{8}{2187} a^{11} + \frac{8}{6561} a^{10} - \frac{4}{729} a^{9} + \frac{20}{6561} a^{8} - \frac{55}{2187} a^{7} - \frac{178}{6561} a^{6} + \frac{286}{2187} a^{5} + \frac{1}{81} a^{4} + \frac{29}{243} a^{3} + \frac{1}{81} a^{2} - \frac{2}{9} a$, $\frac{1}{59049} a^{17} - \frac{1}{59049} a^{16} + \frac{1}{59049} a^{15} - \frac{40}{59049} a^{14} + \frac{16}{59049} a^{13} - \frac{73}{59049} a^{12} + \frac{275}{59049} a^{11} + \frac{100}{59049} a^{10} - \frac{109}{59049} a^{9} + \frac{3145}{59049} a^{8} - \frac{1114}{59049} a^{7} + \frac{955}{59049} a^{6} + \frac{1183}{19683} a^{5} + \frac{545}{6561} a^{4} - \frac{178}{2187} a^{3} - \frac{37}{243} a^{2} + \frac{10}{27} a + \frac{1}{3}$, $\frac{1}{655244256018084775209} a^{18} - \frac{1734582504991952}{218414752006028258403} a^{17} + \frac{11903927845797887}{218414752006028258403} a^{16} + \frac{4104821946032876}{72804917335342752801} a^{15} - \frac{3036186152853113}{72804917335342752801} a^{14} + \frac{83462722858274912}{72804917335342752801} a^{13} - \frac{980622776709598769}{655244256018084775209} a^{12} - \frac{1245455740224743885}{218414752006028258403} a^{11} + \frac{715334737488728995}{218414752006028258403} a^{10} + \frac{273311392052009302}{72804917335342752801} a^{9} - \frac{209081668123268026}{8089435259482528089} a^{8} + \frac{3893357220067461589}{72804917335342752801} a^{7} + \frac{26321453880378688987}{655244256018084775209} a^{6} + \frac{11717714961385447789}{218414752006028258403} a^{5} + \frac{2452032271826344103}{72804917335342752801} a^{4} - \frac{229178813776635238}{24268305778447584267} a^{3} + \frac{226189551324719921}{898826139942503121} a^{2} + \frac{141049775867651425}{299608713314167707} a - \frac{781210329113844}{11096619011635841}$, $\frac{1}{43924422040655493610009079671367669698848399201} a^{19} + \frac{2216849733193348434047915}{14641474013551831203336359890455889899616133067} a^{18} + \frac{88209881868045020367113664371839694630707}{43924422040655493610009079671367669698848399201} a^{17} - \frac{100372714024410512654795851464856415036152}{43924422040655493610009079671367669698848399201} a^{16} + \frac{1066336692497124223132098438595906751194204}{43924422040655493610009079671367669698848399201} a^{15} + \frac{698487531122011015516598063457674736382907}{43924422040655493610009079671367669698848399201} a^{14} - \frac{89933030293408151440172937554473347182384440}{43924422040655493610009079671367669698848399201} a^{13} + \frac{24287864268694439095709495375833986358560821}{43924422040655493610009079671367669698848399201} a^{12} + \frac{56591972668797840435324361009749392737201310}{43924422040655493610009079671367669698848399201} a^{11} + \frac{194700987319254869489001103536006902314255822}{43924422040655493610009079671367669698848399201} a^{10} - \frac{92291257495786886722176042395541152435003272}{43924422040655493610009079671367669698848399201} a^{9} + \frac{1845875203520119501915757057371612892019801976}{43924422040655493610009079671367669698848399201} a^{8} - \frac{195241629521229809535167365565568718469102845}{4880491337850610401112119963485296633205377689} a^{7} + \frac{477694578792258063887553927230203467685217887}{43924422040655493610009079671367669698848399201} a^{6} + \frac{1412480931251458898371619232855710386911225511}{14641474013551831203336359890455889899616133067} a^{5} - \frac{576691260870059721903777384477757676997259222}{4880491337850610401112119963485296633205377689} a^{4} + \frac{185892248640682458112423006299991081795634354}{1626830445950203467037373321161765544401792563} a^{3} + \frac{18838299840477941550656882394943151382176191}{180758938438911496337485924573529504933532507} a^{2} - \frac{3065494620643018573365684950691742663432043}{20084326493212388481942880508169944992614723} a + \frac{868497684773303348089657017823842499748792}{2231591832579154275771431167574438332512747}$

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

Class group and class number

$C_{24298398100}$, which has order $24298398100$ (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:  \( 126615231021.771 \) (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

\(\Q(\sqrt{10}) \), 4.0.256000.2, 5.5.2825761.1, 10.10.817656343461990400000.1

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 ${\href{/LocalNumberField/3.1.0.1}{1} }^{20}$ R $20$ $20$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
$41$41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
41.10.8.1$x^{10} - 27101 x^{5} + 418286592$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$