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Properties

Label 20.0.45804964730...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 3^{17}\cdot 5^{12}\cdot 7^{18}\cdot 19^{5}\cdot 41^{10}$
Root discriminant $1358.46$
Ramified primes $2, 3, 5, 7, 19, 41$
Class number Not computed
Class group Not computed
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![646041051363954996, -4016545382792880, -20343885662967660, 743631596719200, 1142620853270760, 15198620237892, -23930956951560, -981609150060, 701785854420, -2189439000, -10076499167, 117508860, 192372005, -28980, -1799750, -504, 23530, 0, -115, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 115*x^18 + 23530*x^16 - 504*x^15 - 1799750*x^14 - 28980*x^13 + 192372005*x^12 + 117508860*x^11 - 10076499167*x^10 - 2189439000*x^9 + 701785854420*x^8 - 981609150060*x^7 - 23930956951560*x^6 + 15198620237892*x^5 + 1142620853270760*x^4 + 743631596719200*x^3 - 20343885662967660*x^2 - 4016545382792880*x + 646041051363954996)
 
gp: K = bnfinit(x^20 - 115*x^18 + 23530*x^16 - 504*x^15 - 1799750*x^14 - 28980*x^13 + 192372005*x^12 + 117508860*x^11 - 10076499167*x^10 - 2189439000*x^9 + 701785854420*x^8 - 981609150060*x^7 - 23930956951560*x^6 + 15198620237892*x^5 + 1142620853270760*x^4 + 743631596719200*x^3 - 20343885662967660*x^2 - 4016545382792880*x + 646041051363954996, 1)
 

Normalized defining polynomial

\( x^{20} - 115 x^{18} + 23530 x^{16} - 504 x^{15} - 1799750 x^{14} - 28980 x^{13} + 192372005 x^{12} + 117508860 x^{11} - 10076499167 x^{10} - 2189439000 x^{9} + 701785854420 x^{8} - 981609150060 x^{7} - 23930956951560 x^{6} + 15198620237892 x^{5} + 1142620853270760 x^{4} + 743631596719200 x^{3} - 20343885662967660 x^{2} - 4016545382792880 x + 646041051363954996 \)

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:  \(458049647307360970568988220106185818770836600455168000000000000=2^{28}\cdot 3^{17}\cdot 5^{12}\cdot 7^{18}\cdot 19^{5}\cdot 41^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1358.46$
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:  $2, 3, 5, 7, 19, 41$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{294} a^{10} - \frac{1}{6} a^{9} + \frac{19}{49} a^{8} - \frac{41}{98} a^{6} - \frac{5}{14} a^{5} - \frac{40}{147} a^{4} - \frac{5}{21} a^{3} - \frac{5}{49} a^{2} - \frac{2}{7} a - \frac{13}{49}$, $\frac{1}{294} a^{11} - \frac{11}{98} a^{9} - \frac{41}{98} a^{7} + \frac{1}{7} a^{6} + \frac{67}{294} a^{5} + \frac{3}{7} a^{4} - \frac{5}{49} a^{3} - \frac{2}{7} a^{2} - \frac{13}{49} a$, $\frac{1}{294} a^{12} - \frac{1}{6} a^{9} + \frac{37}{98} a^{8} + \frac{1}{7} a^{7} + \frac{62}{147} a^{6} - \frac{5}{14} a^{5} - \frac{4}{49} a^{4} + \frac{4}{21} a^{3} + \frac{18}{49} a^{2} - \frac{3}{7} a + \frac{12}{49}$, $\frac{1}{294} a^{13} - \frac{6}{49} a^{9} + \frac{1}{7} a^{8} + \frac{62}{147} a^{7} + \frac{1}{7} a^{6} + \frac{41}{98} a^{5} - \frac{1}{7} a^{4} + \frac{18}{49} a^{3} - \frac{3}{7} a^{2} + \frac{12}{49} a$, $\frac{1}{588} a^{14} - \frac{1}{588} a^{13} - \frac{5}{147} a^{9} + \frac{5}{42} a^{8} + \frac{53}{147} a^{7} + \frac{3}{28} a^{6} + \frac{57}{196} a^{5} - \frac{1}{7} a^{4} + \frac{22}{147} a^{3} + \frac{23}{98} a - \frac{27}{98}$, $\frac{1}{7056} a^{15} + \frac{5}{7056} a^{13} + \frac{5}{3528} a^{11} + \frac{1}{1176} a^{10} - \frac{199}{3528} a^{9} - \frac{67}{392} a^{8} + \frac{3317}{7056} a^{7} + \frac{9}{98} a^{6} - \frac{2411}{7056} a^{5} - \frac{187}{588} a^{4} - \frac{37}{147} a^{3} - \frac{59}{392} a^{2} - \frac{11}{98} a - \frac{187}{392}$, $\frac{1}{35280} a^{16} - \frac{1}{35280} a^{15} - \frac{1}{5040} a^{14} - \frac{17}{35280} a^{13} + \frac{17}{17640} a^{12} - \frac{13}{8820} a^{11} - \frac{11}{8820} a^{10} - \frac{647}{4410} a^{9} + \frac{4043}{35280} a^{8} + \frac{2567}{7056} a^{7} + \frac{1661}{7056} a^{6} + \frac{1615}{7056} a^{5} + \frac{457}{980} a^{4} + \frac{421}{1960} a^{3} - \frac{17}{40} a^{2} + \frac{381}{1960} a + \frac{1}{280}$, $\frac{1}{35280} a^{17} + \frac{1}{17640} a^{15} - \frac{1}{1470} a^{14} - \frac{53}{35280} a^{13} - \frac{1}{1960} a^{12} + \frac{1}{8820} a^{11} + \frac{5087}{35280} a^{9} - \frac{1957}{5880} a^{8} + \frac{415}{3528} a^{7} - \frac{95}{196} a^{6} + \frac{397}{1680} a^{5} + \frac{191}{392} a^{4} + \frac{1}{70} a^{3} - \frac{481}{980} a^{2} + \frac{31}{70} a - \frac{703}{1960}$, $\frac{1}{93266054898841444321697810362032377403519120} a^{18} - \frac{10362027713124680760484683955753721813}{1480413569822880068598377942254482181008240} a^{17} - \frac{96742973358337092027586925706182988443}{46633027449420722160848905181016188701759560} a^{16} + \frac{50863162271272314310629349615081961479}{1110310177367160051448783456690861635756180} a^{15} - \frac{38894972237827860792084517859490659948381}{93266054898841444321697810362032377403519120} a^{14} - \frac{143893526740796258957958980846147048303}{126892591699104005879860966478955615514992} a^{13} + \frac{1167063276693129110899566441550862748533}{951694437743280044098957248592167116362440} a^{12} + \frac{58996693888489746128644534007395084129}{111031017736716005144878345669086163575618} a^{11} - \frac{14747126598296205776240731749559474430803}{13323722128405920617385401480290339629074160} a^{10} - \frac{335138985325836680463608510124059936230067}{4441240709468640205795133826763446543024720} a^{9} + \frac{26855140885304663496875390696624994640399}{475847218871640022049478624296083558181220} a^{8} - \frac{88825704359370317814313156298338506380075}{222062035473432010289756691338172327151236} a^{7} + \frac{177175611236516088777983761542386855519419}{1480413569822880068598377942254482181008240} a^{6} - \frac{639955899123437663288019846286498189249343}{4441240709468640205795133826763446543024720} a^{5} + \frac{118562756794951015975353141774356573366191}{5181447494380080240094322797890687633528840} a^{4} - \frac{87019273248209409599552160726249305895013}{370103392455720017149594485563620545252060} a^{3} - \frac{10172726100615367682329947763711170830449}{86357458239668004001572046631511460558814} a^{2} + \frac{92745463505994300254644222273214245873457}{246735594970480011433062990375747030168040} a + \frac{428841053009719564689374683785820536569317}{1727149164793360080031440932630229211176280}$, $\frac{1}{50968316330960062025348462734163051257540869879785135514834775302224725869148456035360} a^{19} - \frac{5942105288623546808314188034009569896519}{5096831633096006202534846273416305125754086987978513551483477530222472586914845603536} a^{18} + \frac{52702206901718548686769992788088724376286941411406751107279864923834376282001747}{10193663266192012405069692546832610251508173975957027102966955060444945173829691207072} a^{17} + \frac{8904266920304933400481188558579121300139879802134492039783932294365470922843511}{25484158165480031012674231367081525628770434939892567757417387651112362934574228017680} a^{16} + \frac{1046236026470634110161933225373825889357234469394323922269795353653404604917409891}{25484158165480031012674231367081525628770434939892567757417387651112362934574228017680} a^{15} - \frac{4835542700893875775244478589912475306009298754017539342730719622227430508483965841}{25484158165480031012674231367081525628770434939892567757417387651112362934574228017680} a^{14} - \frac{1367885284794600249119092442670523499455451096344345727973458664841926679937376779}{1820297011820002215191016526220108973483602495706611982672670546508025923898159144120} a^{13} - \frac{825755442242847758876707586956113119763409494449258740021342559436455912399643497}{728118804728000886076406610488043589393440998282644793069068218603210369559263657648} a^{12} + \frac{12038313510876549544491142977795326169815904423676106267459948846967013615748116031}{7281188047280008860764066104880435893934409982826447930690682186032103695592636576480} a^{11} - \frac{2231778865929126077474712399026149425693111559461565165644234341627389874157889889}{3640594023640004430382033052440217946967204991413223965345341093016051847796318288240} a^{10} - \frac{27861269151090875138255110273158741591206522009495243796616350285435821329171800487}{1456237609456001772152813220976087178786881996565289586138136437206420739118527315296} a^{9} + \frac{1279062071665546773120083052751441192125712693535968798541754173740772860760548608773}{3640594023640004430382033052440217946967204991413223965345341093016051847796318288240} a^{8} + \frac{28208939450175313250575592164674407323495427451846472465869606643994915722028394463}{606765670606667405063672175406702991161200831902203994224223515502675307966053048040} a^{7} + \frac{32726110206069454354096229301330179865489071515947977764561608436165235138936636699}{242706268242666962025468870162681196464480332760881597689689406201070123186421219216} a^{6} + \frac{6322267966019243439655038910312706824542348664027495694828805448682780063271921283}{566314625899556244726094030379589458417120776442057061275941947802496954101649511504} a^{5} + \frac{467063065481805410856265531765175895379506436441819509230796426872799476092760362067}{2831573129497781223630470151897947292085603882210285306379709739012484770508247557520} a^{4} + \frac{330639510180842371730354131370707170207006024854838809574214044575362595404433443319}{1415786564748890611815235075948973646042801941105142653189854869506242385254123778760} a^{3} - \frac{46086688940593349996454263738017373625985058915053114659154715586150945884080489781}{235964427458148435302539179324828941007133656850857108864975811584373730875687296460} a^{2} - \frac{10332894997179524641669677778316401216610336844346991666932494832804386552629120739}{47192885491629687060507835864965788201426731370171421772995162316874746175137459292} a - \frac{169948528339613517856762422123707702269116418533099397656715996479259384206914107877}{471928854916296870605078358649657882014267313701714217729951623168747461751374592920}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-287}) \), 4.0.75120528.2, 5.1.388962000.3, 10.0.122696358302636931708000000.1

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 R R $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.9.2$x^{10} + 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed
$7$7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
7.10.9.1$x^{10} - 7$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$41$41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$
41.2.1.2$x^{2} + 246$$2$$1$$1$$C_2$$[\ ]_{2}$