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Properties

Label 20.4.24464628993...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{38}\cdot 3^{16}\cdot 5^{20}\cdot 7^{17}\cdot 31^{5}\cdot 71^{10}$
Root discriminant $4671.18$
Ramified primes $2, 3, 5, 7, 31, 71$
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![7549200162029899, 5606298645781020, -6587629801123320, -11880315149111640, -8369601052982755, -3541108213757136, -974654325915120, -167452964989560, -12907142561690, 1258826067720, 429438591000, 34935460920, -1791460430, -508796400, -18923040, 2452248, 200975, -4260, -720, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 720*x^18 - 4260*x^17 + 200975*x^16 + 2452248*x^15 - 18923040*x^14 - 508796400*x^13 - 1791460430*x^12 + 34935460920*x^11 + 429438591000*x^10 + 1258826067720*x^9 - 12907142561690*x^8 - 167452964989560*x^7 - 974654325915120*x^6 - 3541108213757136*x^5 - 8369601052982755*x^4 - 11880315149111640*x^3 - 6587629801123320*x^2 + 5606298645781020*x + 7549200162029899)
 
gp: K = bnfinit(x^20 - 720*x^18 - 4260*x^17 + 200975*x^16 + 2452248*x^15 - 18923040*x^14 - 508796400*x^13 - 1791460430*x^12 + 34935460920*x^11 + 429438591000*x^10 + 1258826067720*x^9 - 12907142561690*x^8 - 167452964989560*x^7 - 974654325915120*x^6 - 3541108213757136*x^5 - 8369601052982755*x^4 - 11880315149111640*x^3 - 6587629801123320*x^2 + 5606298645781020*x + 7549200162029899, 1)
 

Normalized defining polynomial

\( x^{20} - 720 x^{18} - 4260 x^{17} + 200975 x^{16} + 2452248 x^{15} - 18923040 x^{14} - 508796400 x^{13} - 1791460430 x^{12} + 34935460920 x^{11} + 429438591000 x^{10} + 1258826067720 x^{9} - 12907142561690 x^{8} - 167452964989560 x^{7} - 974654325915120 x^{6} - 3541108213757136 x^{5} - 8369601052982755 x^{4} - 11880315149111640 x^{3} - 6587629801123320 x^{2} + 5606298645781020 x + 7549200162029899 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24464628993938796064014979732395474118735500796153036800000000000000000000=2^{38}\cdot 3^{16}\cdot 5^{20}\cdot 7^{17}\cdot 31^{5}\cdot 71^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $4671.18$
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:  $2, 3, 5, 7, 31, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{4} + \frac{1}{6}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{8} + \frac{1}{6} a^{5} + \frac{1}{3} a^{4} + \frac{1}{12} a + \frac{5}{12}$, $\frac{1}{12} a^{10} - \frac{1}{12} a^{8} + \frac{1}{6} a^{6} - \frac{1}{6} a^{4} + \frac{1}{12} a^{2} - \frac{1}{12}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{8} + \frac{1}{6} a^{7} + \frac{1}{3} a^{4} + \frac{1}{12} a^{3} + \frac{5}{12}$, $\frac{1}{24} a^{12} + \frac{1}{24} a^{8} + \frac{11}{24} a^{4} - \frac{1}{2} a^{2} - \frac{1}{24}$, $\frac{1}{24} a^{13} - \frac{1}{24} a^{9} - \frac{1}{12} a^{8} - \frac{5}{24} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} + \frac{3}{8} a - \frac{1}{12}$, $\frac{1}{24} a^{14} - \frac{1}{24} a^{10} - \frac{1}{12} a^{8} - \frac{5}{24} a^{6} - \frac{1}{6} a^{4} + \frac{3}{8} a^{2} + \frac{5}{12}$, $\frac{1}{672} a^{15} - \frac{1}{224} a^{14} - \frac{3}{224} a^{13} + \frac{1}{96} a^{12} - \frac{1}{96} a^{11} - \frac{3}{224} a^{10} - \frac{1}{32} a^{9} - \frac{1}{224} a^{8} - \frac{101}{672} a^{7} - \frac{31}{224} a^{6} - \frac{53}{224} a^{5} - \frac{17}{224} a^{4} + \frac{1}{224} a^{3} - \frac{37}{224} a^{2} - \frac{15}{32} a + \frac{319}{672}$, $\frac{1}{4032} a^{16} - \frac{1}{224} a^{14} + \frac{1}{112} a^{13} + \frac{5}{288} a^{12} + \frac{23}{672} a^{11} + \frac{5}{168} a^{10} - \frac{11}{672} a^{9} - \frac{37}{672} a^{8} + \frac{23}{336} a^{7} - \frac{17}{672} a^{6} - \frac{1}{168} a^{5} + \frac{233}{2016} a^{4} - \frac{157}{672} a^{3} - \frac{19}{42} a^{2} + \frac{325}{672} a + \frac{397}{4032}$, $\frac{1}{4032} a^{17} - \frac{1}{224} a^{14} + \frac{19}{1008} a^{13} - \frac{1}{56} a^{12} - \frac{1}{672} a^{11} + \frac{3}{112} a^{10} - \frac{1}{42} a^{9} - \frac{19}{672} a^{8} + \frac{1}{42} a^{7} + \frac{55}{224} a^{6} + \frac{31}{1008} a^{5} + \frac{41}{336} a^{4} - \frac{295}{672} a^{3} - \frac{3}{7} a^{2} - \frac{1073}{4032} a + \frac{5}{672}$, $\frac{1}{4032} a^{18} + \frac{11}{2016} a^{14} - \frac{11}{672} a^{13} - \frac{1}{84} a^{12} - \frac{1}{224} a^{11} + \frac{13}{672} a^{10} + \frac{1}{336} a^{9} - \frac{1}{32} a^{8} - \frac{23}{112} a^{7} - \frac{439}{2016} a^{6} + \frac{25}{672} a^{5} + \frac{3}{8} a^{4} + \frac{19}{224} a^{3} - \frac{719}{4032} a^{2} - \frac{5}{14} a - \frac{23}{672}$, $\frac{1}{72248846285514751595969030349737327680903759472408253138888673615699504148233181299242125238855314313805871132736} a^{19} + \frac{6156738605404590022633507942041934251100935269423869084082715779543108872097703083009514299353597303200773363}{72248846285514751595969030349737327680903759472408253138888673615699504148233181299242125238855314313805871132736} a^{18} - \frac{106347685602590230469460734583516431215664277159416584966727671054281324419858219950370906963734326068684137}{9031105785689343949496128793717165960112969934051031642361084201962438018529147662405265654856914289225733891592} a^{17} - \frac{34466257246774813495209021812417916300745632982785027465435302695384645320255744039476580335430091348917299}{1290157969384191992785161256245309422873281419150147377480154885994634002647021094629323664979559184175104841656} a^{16} + \frac{6288419591365193774264609139306163746589367165228887509639099503957174023841519484411471008778503970858105437}{9031105785689343949496128793717165960112969934051031642361084201962438018529147662405265654856914289225733891592} a^{15} - \frac{578000474609422543422305435230074038801426710013584972682786384109795443342624905534148810671005693960573919727}{36124423142757375797984515174868663840451879736204126569444336807849752074116590649621062619427657156902935566368} a^{14} + \frac{20618393127906252734732738575657540842366534440292090181617564120041934269387158823004927282619608750080398607}{2257776446422335987374032198429291490028242483512757910590271050490609504632286915601316413714228572306433472898} a^{13} + \frac{108820725370227377606199282600168656288724379701331331891389905980399793160296989072977881752879267338486338443}{18062211571378687898992257587434331920225939868102063284722168403924876037058295324810531309713828578451467783184} a^{12} - \frac{20746520187686086882083498819622257489424924848896654637418634696180075484003736098044994374187801079084621741}{573403541948529774571182780553470854610347297400065501102291060442059556732009375390810517768692970744491040736} a^{11} - \frac{195006872247017981075373445446031592579794541941473458084944379819926337075462150017585012716167179791127848719}{6020737190459562632997419195811443973408646622700687761574056134641625345686098441603510436571276192817155927728} a^{10} - \frac{8975501500213485812501827852049520991468944735783941006590639829510520474951813283204050247322939915622946977}{286701770974264887285591390276735427305173648700032750551145530221029778366004687695405258884346485372245520368} a^{9} - \frac{139356678141803762294185405852771950809595060332504748520919954410477414100343413882353518631289604471394660711}{2006912396819854210999139731937147991136215540900229253858018711547208448562032813867836812190425397605718642576} a^{8} - \frac{36189264766418088794813362314896931203775123280591832835768477732324292984714526496776040514432795035533005079}{368616562681197712224331787498659835106651834042899250708615681712752572184863169894092475708445481192887097616} a^{7} - \frac{3316398241331185235538773079701866025703848240845418467467761834689021700517683980565079498458972986522146788699}{36124423142757375797984515174868663840451879736204126569444336807849752074116590649621062619427657156902935566368} a^{6} - \frac{4352121228477476459910951854349131024056501567109934073490128961198633064790454864178984095281697150478959384159}{18062211571378687898992257587434331920225939868102063284722168403924876037058295324810531309713828578451467783184} a^{5} - \frac{2393209844729699411662364843662053788328782923020241303457067717908595412010136079972734887746891780668498383747}{9031105785689343949496128793717165960112969934051031642361084201962438018529147662405265654856914289225733891592} a^{4} - \frac{4400297072502011886418503565918050008312414282129247593934979734904739679424868517607274387358417738454988917745}{10321263755073535942281290049962475382986251353201179019841239087957072021176168757034589319836473473400838733248} a^{3} - \frac{3080165649907449048814193705111327899333503229826597545718777254549321934295265198982832851799164271465343264549}{10321263755073535942281290049962475382986251353201179019841239087957072021176168757034589319836473473400838733248} a^{2} - \frac{300590044169962188099816882288528726385431946097703851572657689345284599778715688404657919109670977915540177294}{1128888223211167993687016099214645745014121241756378955295135525245304752316143457800658206857114286153216736449} a + \frac{2872217169694844839984832649538454930926552307191726281728328461983986141216696798000638722974703029888844137709}{9031105785689343949496128793717165960112969934051031642361084201962438018529147662405265654856914289225733891592}$

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:  $11$
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

\(\Q(\sqrt{142}) \), 4.4.70009408.1, 5.1.9724050000.14, 10.2.349394498549049880120320000000000.1

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 $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$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$$[\ ]_{10}^{4}$
7.10.8.1$x^{10} - 7 x^{5} + 147$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$31$31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
31.2.1.2$x^{2} + 217$$2$$1$$1$$C_2$$[\ ]_{2}$
$71$71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$
71.2.1.1$x^{2} - 71$$2$$1$$1$$C_2$$[\ ]_{2}$