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Properties

Label 20.0.43603533371...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{17}\cdot 5^{15}\cdot 7^{18}\cdot 11^{17}\cdot 29^{5}$
Root discriminant $1520.46$
Ramified primes $2, 3, 5, 7, 11, 29$
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![12360603018214069136, -340409838530166384, 95248235112201248, -97626382174939856, 14114690556587824, 713448227711744, 705392864018584, 15064947579416, 12983556569844, 419389149256, 173113722695, -2196573753, 1327351367, -8835250, 8522143, -695863, 49105, 258, -99, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 99*x^18 + 258*x^17 + 49105*x^16 - 695863*x^15 + 8522143*x^14 - 8835250*x^13 + 1327351367*x^12 - 2196573753*x^11 + 173113722695*x^10 + 419389149256*x^9 + 12983556569844*x^8 + 15064947579416*x^7 + 705392864018584*x^6 + 713448227711744*x^5 + 14114690556587824*x^4 - 97626382174939856*x^3 + 95248235112201248*x^2 - 340409838530166384*x + 12360603018214069136)
 
gp: K = bnfinit(x^20 - 5*x^19 - 99*x^18 + 258*x^17 + 49105*x^16 - 695863*x^15 + 8522143*x^14 - 8835250*x^13 + 1327351367*x^12 - 2196573753*x^11 + 173113722695*x^10 + 419389149256*x^9 + 12983556569844*x^8 + 15064947579416*x^7 + 705392864018584*x^6 + 713448227711744*x^5 + 14114690556587824*x^4 - 97626382174939856*x^3 + 95248235112201248*x^2 - 340409838530166384*x + 12360603018214069136, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 99 x^{18} + 258 x^{17} + 49105 x^{16} - 695863 x^{15} + 8522143 x^{14} - 8835250 x^{13} + 1327351367 x^{12} - 2196573753 x^{11} + 173113722695 x^{10} + 419389149256 x^{9} + 12983556569844 x^{8} + 15064947579416 x^{7} + 705392864018584 x^{6} + 713448227711744 x^{5} + 14114690556587824 x^{4} - 97626382174939856 x^{3} + 95248235112201248 x^{2} - 340409838530166384 x + 12360603018214069136 \)

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:  \(4360353337131964370592434786665009884862483930746000000000000000=2^{16}\cdot 3^{17}\cdot 5^{15}\cdot 7^{18}\cdot 11^{17}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1520.46$
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:  $2, 3, 5, 7, 11, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{22} a^{9} + \frac{9}{22} a^{8} + \frac{1}{11} a^{7} - \frac{7}{22} a^{6} - \frac{5}{11} a^{5} + \frac{7}{22} a^{4} - \frac{3}{22} a^{3} - \frac{1}{11} a^{2} + \frac{5}{11} a + \frac{4}{11}$, $\frac{1}{22} a^{10} + \frac{9}{22} a^{8} - \frac{3}{22} a^{7} + \frac{9}{22} a^{6} + \frac{9}{22} a^{5} + \frac{3}{22} a^{3} + \frac{3}{11} a^{2} + \frac{3}{11} a - \frac{3}{11}$, $\frac{1}{22} a^{11} + \frac{2}{11} a^{8} - \frac{9}{22} a^{7} + \frac{3}{11} a^{6} + \frac{1}{11} a^{5} + \frac{3}{11} a^{4} - \frac{1}{2} a^{3} + \frac{1}{11} a^{2} - \frac{4}{11} a - \frac{3}{11}$, $\frac{1}{110} a^{12} + \frac{1}{55} a^{11} - \frac{1}{110} a^{10} - \frac{23}{55} a^{8} + \frac{49}{110} a^{7} + \frac{3}{10} a^{6} + \frac{19}{110} a^{5} - \frac{49}{110} a^{4} + \frac{3}{10} a^{3} - \frac{23}{55} a^{2} + \frac{2}{11} a + \frac{3}{55}$, $\frac{1}{110} a^{13} + \frac{1}{55} a^{10} - \frac{1}{110} a^{9} + \frac{8}{55} a^{8} - \frac{2}{11} a^{7} - \frac{1}{55} a^{6} + \frac{23}{110} a^{5} + \frac{18}{55} a^{4} + \frac{14}{55} a^{3} + \frac{16}{55} a^{2} + \frac{23}{55} a - \frac{6}{55}$, $\frac{1}{110} a^{14} + \frac{1}{55} a^{11} - \frac{1}{110} a^{10} + \frac{1}{110} a^{9} - \frac{9}{22} a^{8} - \frac{16}{55} a^{7} + \frac{9}{55} a^{6} - \frac{17}{55} a^{5} + \frac{3}{10} a^{4} - \frac{3}{10} a^{3} - \frac{17}{55} a^{2} - \frac{26}{55} a - \frac{1}{11}$, $\frac{1}{14520} a^{15} - \frac{17}{4840} a^{14} - \frac{5}{2904} a^{13} + \frac{1}{330} a^{12} + \frac{1}{14520} a^{11} + \frac{19}{2904} a^{10} + \frac{329}{14520} a^{9} + \frac{63}{1210} a^{8} - \frac{1297}{14520} a^{7} + \frac{223}{4840} a^{6} - \frac{1075}{2904} a^{5} + \frac{2423}{7260} a^{4} - \frac{3}{22} a^{3} - \frac{259}{726} a^{2} + \frac{364}{1815} a + \frac{1273}{3630}$, $\frac{1}{12211320} a^{16} - \frac{1}{3052830} a^{15} - \frac{89}{6105660} a^{14} + \frac{14803}{4070440} a^{13} + \frac{23717}{12211320} a^{12} - \frac{31807}{6105660} a^{11} + \frac{6387}{2035220} a^{10} + \frac{174223}{12211320} a^{9} + \frac{2717927}{12211320} a^{8} + \frac{2374367}{6105660} a^{7} + \frac{1246921}{3052830} a^{6} - \frac{1881401}{4070440} a^{5} - \frac{1935089}{6105660} a^{4} - \frac{56071}{1526415} a^{3} + \frac{4937}{277530} a^{2} + \frac{1248767}{3052830} a - \frac{1440547}{3052830}$, $\frac{1}{12211320} a^{17} - \frac{97}{6105660} a^{15} + \frac{43697}{12211320} a^{14} - \frac{20671}{12211320} a^{13} + \frac{5209}{2035220} a^{12} - \frac{108067}{6105660} a^{11} - \frac{116537}{12211320} a^{10} - \frac{82859}{4070440} a^{9} + \frac{253653}{2035220} a^{8} - \frac{188519}{610566} a^{7} + \frac{5314561}{12211320} a^{6} - \frac{2455331}{6105660} a^{5} - \frac{189567}{1017610} a^{4} + \frac{482818}{1526415} a^{3} - \frac{473439}{1017610} a^{2} + \frac{89237}{610566} a - \frac{327818}{1526415}$, $\frac{1}{1475967910173282034278685566920754521306160} a^{18} - \frac{44537625280713549444389300291378983}{1475967910173282034278685566920754521306160} a^{17} - \frac{8430967930235890287001693111140981}{491989303391094011426228522306918173768720} a^{16} + \frac{57311421002546371440793044936267241}{6361930647298629458097782616037735005630} a^{15} - \frac{2125726242258623856243188959224584591963}{1475967910173282034278685566920754521306160} a^{14} - \frac{102694039062760032454936563438278923105}{26835780184968764259612464853104627660112} a^{13} + \frac{1907471269037958919436818158429644707093}{491989303391094011426228522306918173768720} a^{12} + \frac{215652445309661623044609521071065821176}{92247994385830127142417847932547157581635} a^{11} - \frac{1903856403085575318457180806672367062491}{98397860678218802285245704461383634753744} a^{10} - \frac{1348753028465265166716756095601519386735}{98397860678218802285245704461383634753744} a^{9} - \frac{99591588604109369287529508929348361518147}{491989303391094011426228522306918173768720} a^{8} + \frac{103988121144461509255479901093952598911209}{737983955086641017139342783460377260653080} a^{7} - \frac{9804880808162553843627064243217746147063}{30749331461943375714139282644182385860545} a^{6} - \frac{528081391419033839077346645223079150423}{4472630030828127376602077475517437943352} a^{5} - \frac{159633782878822154387413186984534813221721}{368991977543320508569671391730188630326540} a^{4} - \frac{44448501077471320585915592535378701339799}{122997325847773502856557130576729543442180} a^{3} - \frac{74328403149730082222810031020674229160167}{184495988771660254284835695865094315163270} a^{2} - \frac{26389261997690711437775444448580314902459}{61498662923886751428278565288364771721090} a - \frac{18209961453622968172036664985285647087627}{36899197754332050856967139173018863032654}$, $\frac{1}{112084660027840031664461233000881015049440617165982212992980277159828082367956273907153434009258640} a^{19} + \frac{857815634896005440209239310509628273463477497686256343}{7005291251740001979028827062555063440590038572873888312061267322489255147997267119197089625578665} a^{18} - \frac{558246843358683637129238063715147086448589873418092155026479712063395460703098773460016017}{14010582503480003958057654125110126881180077145747776624122534644978510295994534238394179251157330} a^{17} + \frac{4271157042585673872619255097776658918570841281492908983581941244650695337930257729433586339}{112084660027840031664461233000881015049440617165982212992980277159828082367956273907153434009258640} a^{16} - \frac{21264258381911666146927272405919595154857839861618367287619724043484336448120947442902972797}{2490770222840889592543582955575133667765347048132938066510672825773957385954583864603409644650192} a^{15} - \frac{6999163979971221492603402555178614934988943032068891170742026192993821831946590039421309777247}{5094757273992728712020965136403682502247300780271918772408194416355821925816194268506974273148120} a^{14} - \frac{205567371588765336591169937949508789270790054501795235658134533602387926988109761023959544794}{778365694637777997669869673617229271176670952541543145784585258054361683110807457688565513953185} a^{13} - \frac{2755007283527825408410568480794264822170957397968295480420745175655268435219340992035297868587}{772997655364414011479042986212972517582349083903325606848139842481572981847974302807954717305232} a^{12} + \frac{871553621541236824227491023346861703903311748484516680752372213258493407401040960183578671232069}{112084660027840031664461233000881015049440617165982212992980277159828082367956273907153434009258640} a^{11} - \frac{893462400657873682792134898970011137608168964381399126391603542109730701554653544829021646095723}{56042330013920015832230616500440507524720308582991106496490138579914041183978136953576717004629320} a^{10} + \frac{113060723047724672295724466837511710860510485882120609481002325429949624397052303651234709164291}{6226925557102223981358957388937834169413367620332345166276682064434893464886459661508524111625480} a^{9} + \frac{932333326059896807316214631609775272989910640000236270407898343172567279076184144599643740368283}{3864988276822070057395214931064862587911745419516628034240699212407864909239871514039773586526160} a^{8} - \frac{4819402651393073093245079236268672734334370001452196782372687782873519167695546639475451103259}{966247069205517514348803732766215646977936354879157008560174803101966227309967878509943396631540} a^{7} + \frac{174322868730213777668074810010486963822149973618298146751584703992357575597243167334454912439409}{849126212332121452003494189400613750374550130045319795401365736059303654302699044751162378858020} a^{6} - \frac{102036231855057333565007105777350162625116783362313927020761958219202861767281609558190955290652}{1401058250348000395805765412511012688118007714574777662412253464497851029599453423839417925115733} a^{5} + \frac{615035204012026160341592687324302022770549684666686003223427143264006740154414864444663111425401}{1868077667130667194407687216681350250824010286099703549883004619330468039465937898452557233487644} a^{4} - \frac{3348398270507814779127271543116114953933573913055606661369717568024917789284000258319771317065719}{28021165006960007916115308250220253762360154291495553248245069289957020591989068476788358502314660} a^{3} - \frac{6993534877405670530146645918398483633282508617078799738900807231142627002630143349338177481630041}{14010582503480003958057654125110126881180077145747776624122534644978510295994534238394179251157330} a^{2} + \frac{3466862791414904307754232212043502548757921927744298266168145443846117577929356235770932284903121}{7005291251740001979028827062555063440590038572873888312061267322489255147997267119197089625578665} a + \frac{2272170551159015380268304061369571255464655222519271380694493973848770619352530046454513582913}{57894969022644644454783694731860028434628417957635440595547663822225249157002207596670162194865}$

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

\(\Q(\sqrt{-35}) \), 4.0.5861625.3, 5.1.5694792642000.5, 10.0.1135073213238206905740000000.1

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 R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
2.10.8.1$x^{10} - 2 x^{5} + 4$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$3$3.10.9.1$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}$
$11$11.10.9.3$x^{10} - 891$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.8.2$x^{10} + 143 x^{5} + 5929$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$