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Properties

Label 20.0.73445842291...4192.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{10}\cdot 11^{13}$
Root discriminant $55.37$
Ramified primes $2, 3, 11$
Class number $4$ (GRH)
Class group $[2, 2]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![279487159291, -49255787044, -74360770750, 3758832104, 18407809245, -1486736436, -2308326270, 66920760, 273021994, -7617672, -20955572, -403332, 1404639, 47192, -63982, -4352, 2405, 144, -54, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 54*x^18 + 144*x^17 + 2405*x^16 - 4352*x^15 - 63982*x^14 + 47192*x^13 + 1404639*x^12 - 403332*x^11 - 20955572*x^10 - 7617672*x^9 + 273021994*x^8 + 66920760*x^7 - 2308326270*x^6 - 1486736436*x^5 + 18407809245*x^4 + 3758832104*x^3 - 74360770750*x^2 - 49255787044*x + 279487159291)
 
gp: K = bnfinit(x^20 - 4*x^19 - 54*x^18 + 144*x^17 + 2405*x^16 - 4352*x^15 - 63982*x^14 + 47192*x^13 + 1404639*x^12 - 403332*x^11 - 20955572*x^10 - 7617672*x^9 + 273021994*x^8 + 66920760*x^7 - 2308326270*x^6 - 1486736436*x^5 + 18407809245*x^4 + 3758832104*x^3 - 74360770750*x^2 - 49255787044*x + 279487159291, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 54 x^{18} + 144 x^{17} + 2405 x^{16} - 4352 x^{15} - 63982 x^{14} + 47192 x^{13} + 1404639 x^{12} - 403332 x^{11} - 20955572 x^{10} - 7617672 x^{9} + 273021994 x^{8} + 66920760 x^{7} - 2308326270 x^{6} - 1486736436 x^{5} + 18407809245 x^{4} + 3758832104 x^{3} - 74360770750 x^{2} - 49255787044 x + 279487159291 \)

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:  \(73445842291921443780364922039304192=2^{55}\cdot 3^{10}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.37$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} + \frac{8}{19} a^{16} + \frac{5}{19} a^{15} - \frac{5}{19} a^{14} + \frac{4}{19} a^{13} + \frac{1}{19} a^{12} + \frac{1}{19} a^{11} - \frac{3}{19} a^{10} - \frac{7}{19} a^{9} - \frac{2}{19} a^{8} - \frac{7}{19} a^{7} - \frac{7}{19} a^{6} - \frac{1}{19} a^{5} - \frac{7}{19} a^{4} + \frac{3}{19} a^{3} + \frac{6}{19} a^{2} - \frac{6}{19} a$, $\frac{1}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{19} + \frac{690891565004301888018773684231486516055273473785876229876897311131666681823385762869290489681662916}{52294384289192903989781281467985771888387109518723803677727875730924565179874281888916784223889584697} a^{18} + \frac{279033145910311516316119408890403415292960446471243576091949786191987299678601288884019012667366244472}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{17} + \frac{13875759091222064846530727531072179291570255839220786190081393993222244167506654467686665604106745549}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{16} + \frac{322536034669107025645318055646454741514445078707298553045203120148750736805007109706480793392107303851}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{15} + \frac{10398131375047393309212221393814024837255619295980507595415943233896235608091948111384923931111258107}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{14} + \frac{196781623195519901481037536537942457685942740495898461766955863735634087800252757145213348849789980031}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{13} - \frac{35994398938594310105004557135168318539231551626955174657523733966069689482083265203274515679166830757}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{12} - \frac{274872464278529536758021065401456402419645045246586216641327627410250112126746869039170854731607044541}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{11} + \frac{102689955954365177360860578705229564602686023441051293863423674439737725591552038348894885138559987479}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{10} - \frac{333487820452451847692335979700301796850163250686653006812490835879488170481172870081173512465463813691}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{9} + \frac{271855093415161128256112813115340391099459846284814298295309133685559731110722374748478024063090655408}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{8} - \frac{386290795117251597433377232623157791007476514099174631489450936409095192407146157299939002500685611}{5550800567009302658133208647439830535638855200311465194842623680936127030265985228432507822647497817} a^{7} + \frac{432725512231720721135990526635557296697748008378386425986496195123766159153928753189496508748231449746}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{6} + \frac{235377484830117790013883022987062385557778040683626142022464699670084064260364636472282882747455669674}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{5} + \frac{230674552121014709198609688632196525501659277254341690723838341690099791640766483736902064740785390610}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{4} - \frac{472658028037088839383859106783031251751278818911299772569252121091746983381832819459953090424218326322}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{3} + \frac{56463554444696898835600094218055398902300114403448887155309890945912128279214547150002913071542487881}{993593301494665175805844347891729665879355080855752269876829638887566738417611355889418900253902109243} a^{2} + \frac{24390869482752055673169700730933812962461985677151758593949185674089080800752435952736258339172448578}{52294384289192903989781281467985771888387109518723803677727875730924565179874281888916784223889584697} a - \frac{16481274373904654394736804912399543394879375482313789610974289325062439132215966535799253400306599227}{52294384289192903989781281467985771888387109518723803677727875730924565179874281888916784223889584697}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$ (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:  \( 480489914.07005835 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-2}) \), 4.0.202752.5, 10.0.479756288.1

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

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{10}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$