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Properties

Label 20.0.14368590466...5201.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 1093^{10}$
Root discriminant $57.26$
Ramified primes $3, 1093$
Class number $3146$ (GRH)
Class group $[11, 286]$ (GRH)
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2560000, -5184000, 5809600, -4678560, 2009536, -197640, 184336, -84090, 139181, -101226, 25630, -1920, -1266, 690, 330, -222, 26, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^18 + 26*x^16 - 222*x^15 + 330*x^14 + 690*x^13 - 1266*x^12 - 1920*x^11 + 25630*x^10 - 101226*x^9 + 139181*x^8 - 84090*x^7 + 184336*x^6 - 197640*x^5 + 2009536*x^4 - 4678560*x^3 + 5809600*x^2 - 5184000*x + 2560000)
 
gp: K = bnfinit(x^20 - 6*x^18 + 26*x^16 - 222*x^15 + 330*x^14 + 690*x^13 - 1266*x^12 - 1920*x^11 + 25630*x^10 - 101226*x^9 + 139181*x^8 - 84090*x^7 + 184336*x^6 - 197640*x^5 + 2009536*x^4 - 4678560*x^3 + 5809600*x^2 - 5184000*x + 2560000, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{18} + 26 x^{16} - 222 x^{15} + 330 x^{14} + 690 x^{13} - 1266 x^{12} - 1920 x^{11} + 25630 x^{10} - 101226 x^{9} + 139181 x^{8} - 84090 x^{7} + 184336 x^{6} - 197640 x^{5} + 2009536 x^{4} - 4678560 x^{3} + 5809600 x^{2} - 5184000 x + 2560000 \)

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:  \(143685904663725311070484819161855201=3^{10}\cdot 1093^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.26$
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:  $3, 1093$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{4}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{8} a^{7} + \frac{1}{8} a^{6} + \frac{1}{16} a^{5} + \frac{3}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{12} - \frac{1}{32} a^{11} + \frac{1}{32} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{7}{32} a^{6} - \frac{5}{32} a^{5} - \frac{7}{32} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{160} a^{13} - \frac{1}{160} a^{12} + \frac{3}{160} a^{11} - \frac{1}{20} a^{10} - \frac{1}{40} a^{9} - \frac{7}{80} a^{8} - \frac{19}{160} a^{7} + \frac{7}{160} a^{6} + \frac{19}{160} a^{5} - \frac{17}{80} a^{4} + \frac{29}{80} a^{3} - \frac{3}{40} a^{2} + \frac{3}{20} a$, $\frac{1}{160} a^{14} + \frac{1}{80} a^{12} - \frac{1}{32} a^{11} + \frac{1}{20} a^{10} + \frac{1}{80} a^{9} + \frac{7}{160} a^{8} - \frac{3}{40} a^{7} + \frac{13}{80} a^{6} - \frac{3}{32} a^{5} + \frac{1}{40} a^{4} + \frac{13}{80} a^{3} - \frac{7}{40} a^{2} + \frac{3}{20} a$, $\frac{1}{160} a^{15} + \frac{1}{80} a^{12} - \frac{3}{160} a^{11} + \frac{3}{160} a^{10} + \frac{1}{32} a^{9} - \frac{7}{80} a^{8} - \frac{3}{80} a^{7} + \frac{1}{10} a^{6} + \frac{21}{160} a^{5} - \frac{1}{160} a^{4} - \frac{1}{40} a^{3} + \frac{3}{10} a^{2} - \frac{1}{20} a$, $\frac{1}{640} a^{16} + \frac{1}{640} a^{15} + \frac{1}{640} a^{14} - \frac{1}{640} a^{13} - \frac{1}{640} a^{12} - \frac{9}{640} a^{11} + \frac{7}{128} a^{10} + \frac{3}{128} a^{9} + \frac{19}{640} a^{8} - \frac{15}{128} a^{7} - \frac{123}{640} a^{6} + \frac{13}{640} a^{5} - \frac{1}{10} a^{4} + \frac{29}{160} a^{3} - \frac{17}{40} a^{2} - \frac{1}{40} a$, $\frac{1}{640} a^{17} - \frac{1}{320} a^{14} - \frac{1}{80} a^{12} + \frac{1}{160} a^{11} + \frac{1}{32} a^{10} - \frac{9}{160} a^{9} + \frac{33}{320} a^{8} + \frac{1}{20} a^{7} + \frac{7}{80} a^{6} - \frac{117}{640} a^{5} + \frac{3}{32} a^{4} + \frac{53}{160} a^{3} + \frac{1}{40} a^{2} - \frac{9}{40} a$, $\frac{1}{1134001350400} a^{18} - \frac{14215765}{22680027008} a^{17} - \frac{110341269}{283500337600} a^{16} - \frac{5245829}{22680027008} a^{15} - \frac{885114521}{283500337600} a^{14} - \frac{44438643}{283500337600} a^{13} + \frac{23757541}{2984214080} a^{12} + \frac{150190905}{11340013504} a^{11} + \frac{1269350087}{35437542200} a^{10} - \frac{345580153}{5968428160} a^{9} + \frac{6329508921}{56700067520} a^{8} - \frac{5343875109}{283500337600} a^{7} + \frac{252969278131}{1134001350400} a^{6} - \frac{9943328161}{113400135040} a^{5} - \frac{5004860771}{283500337600} a^{4} - \frac{677676911}{28350033760} a^{3} - \frac{25242950719}{70875084400} a^{2} - \frac{2804814629}{7087508440} a - \frac{28960203}{177187711}$, $\frac{1}{101064672131882127441703699120960000} a^{19} + \frac{894262339844252077227}{10106467213188212744170369912096000} a^{18} + \frac{11166982910446184969580075930947}{50532336065941063720851849560480000} a^{17} - \frac{94066769317529169212519595499}{265959663504952966951851839792000} a^{16} + \frac{126588771156781073740811219881313}{50532336065941063720851849560480000} a^{15} + \frac{131345555664702774748355849566399}{50532336065941063720851849560480000} a^{14} + \frac{21635099436587114411866544796779}{10106467213188212744170369912096000} a^{13} - \frac{101963589977458405810564832297001}{10106467213188212744170369912096000} a^{12} - \frac{302149271254774452619919889493983}{50532336065941063720851849560480000} a^{11} - \frac{155020604492711966548552856196837}{5053233606594106372085184956048000} a^{10} - \frac{57411111839452230220377109981817}{10106467213188212744170369912096000} a^{9} - \frac{3106112475568909272685490118867563}{50532336065941063720851849560480000} a^{8} + \frac{9690593941340436082605990009811161}{101064672131882127441703699120960000} a^{7} - \frac{46241605291782889023793498645031}{5053233606594106372085184956048000} a^{6} - \frac{228362933738437523939268936528627}{3158271004121316482553240597530000} a^{5} - \frac{49584423787692246959352552934181}{315827100412131648255324059753000} a^{4} - \frac{679032702730593615096992881153451}{1579135502060658241276620298765000} a^{3} - \frac{80030473085746241179415648117287}{315827100412131648255324059753000} a^{2} - \frac{1626136372184444535769898644493}{3324495793811912086898147997400} a + \frac{304862790894462280647372193183}{1579135502060658241276620298765}$

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

Class group and class number

$C_{11}\times C_{286}$, which has order $3146$ (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:  \( -\frac{347641195001021805567}{570381950088412889095360000} a^{19} - \frac{14942919787370220921}{14259548752210322227384000} a^{18} + \frac{851467559838550182301}{285190975044206444547680000} a^{17} + \frac{22632679806808001919}{3564887188052580556846000} a^{16} - \frac{3585747220941091674971}{285190975044206444547680000} a^{15} + \frac{29664017753165292214017}{285190975044206444547680000} a^{14} + \frac{1017153819001375471657}{57038195008841288909536000} a^{13} - \frac{33621267633550665842483}{57038195008841288909536000} a^{12} - \frac{62584997554860196341889}{285190975044206444547680000} a^{11} + \frac{7236268977505988158963}{3564887188052580556846000} a^{10} - \frac{692675353828770203285861}{57038195008841288909536000} a^{9} + \frac{10014231248911813310832971}{285190975044206444547680000} a^{8} + \frac{543675208005502505873213}{570381950088412889095360000} a^{7} - \frac{1020534751912049116518221}{57038195008841288909536000} a^{6} - \frac{1120229355695992045088433}{8912217970131451392115000} a^{5} - \frac{609363429218920013711609}{14259548752210322227384000} a^{4} - \frac{4479565034692954898449329}{4456108985065725696057500} a^{3} + \frac{2625684995800135756078383}{3564887188052580556846000} a^{2} - \frac{118935824276283978386981}{356488718805258055684600} a + \frac{7766529071475718711009}{8912217970131451392115} \) (order $6$)
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:  \( 25892885.17 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{1093}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3279}) \), \(\Q(\sqrt{-3}, \sqrt{1093})\), 5.5.1194649.1 x5, 10.10.1559914552888693.1, 10.0.346806254667843.1 x5, 10.0.379059236351952399.1 x5

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

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$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}$
1093Data not computed