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Properties

Label 20.0.10675123826...000.18
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}$
Root discriminant $282.76$
Ramified primes $2, 3, 5, 7$
Class number $890101760$ (GRH)
Class group $[2, 4, 4, 4, 4, 4, 434620]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![256823092541401, 0, 67422649492370, 0, 8125792114845, 0, 595507325160, 0, 29820903730, 0, 1089449212, 0, 30140770, 0, 640520, 0, 10485, 0, 130, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 130*x^18 + 10485*x^16 + 640520*x^14 + 30140770*x^12 + 1089449212*x^10 + 29820903730*x^8 + 595507325160*x^6 + 8125792114845*x^4 + 67422649492370*x^2 + 256823092541401)
 
gp: K = bnfinit(x^20 + 130*x^18 + 10485*x^16 + 640520*x^14 + 30140770*x^12 + 1089449212*x^10 + 29820903730*x^8 + 595507325160*x^6 + 8125792114845*x^4 + 67422649492370*x^2 + 256823092541401, 1)
 

Normalized defining polynomial

\( x^{20} + 130 x^{18} + 10485 x^{16} + 640520 x^{14} + 30140770 x^{12} + 1089449212 x^{10} + 29820903730 x^{8} + 595507325160 x^{6} + 8125792114845 x^{4} + 67422649492370 x^{2} + 256823092541401 \)

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:  \(10675123826048640000000000000000000000000000000000=2^{40}\cdot 3^{10}\cdot 5^{34}\cdot 7^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $282.76$
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:  $2, 3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4200=2^{3}\cdot 3\cdot 5^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{4200}(1,·)$, $\chi_{4200}(3779,·)$, $\chi_{4200}(841,·)$, $\chi_{4200}(589,·)$, $\chi_{4200}(1681,·)$, $\chi_{4200}(1429,·)$, $\chi_{4200}(2521,·)$, $\chi_{4200}(2269,·)$, $\chi_{4200}(671,·)$, $\chi_{4200}(3361,·)$, $\chi_{4200}(419,·)$, $\chi_{4200}(3109,·)$, $\chi_{4200}(1511,·)$, $\chi_{4200}(1259,·)$, $\chi_{4200}(3949,·)$, $\chi_{4200}(2351,·)$, $\chi_{4200}(2099,·)$, $\chi_{4200}(3191,·)$, $\chi_{4200}(2939,·)$, $\chi_{4200}(4031,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{9} - \frac{1}{64} a^{8} + \frac{1}{32} a^{7} - \frac{1}{32} a^{6} + \frac{1}{32} a^{5} - \frac{1}{32} a^{4} - \frac{3}{64} a^{3} + \frac{3}{64} a^{2} + \frac{29}{64} a - \frac{29}{64}$, $\frac{1}{448} a^{12} - \frac{1}{64} a^{8} + \frac{1}{28} a^{6} - \frac{3}{64} a^{4} - \frac{1}{4} a^{2} - \frac{69}{448}$, $\frac{1}{896} a^{13} - \frac{1}{896} a^{12} - \frac{1}{128} a^{9} + \frac{1}{128} a^{8} + \frac{1}{56} a^{7} - \frac{1}{56} a^{6} + \frac{13}{128} a^{5} - \frac{13}{128} a^{4} - \frac{1}{8} a^{3} + \frac{1}{8} a^{2} + \frac{267}{896} a - \frac{267}{896}$, $\frac{1}{6272} a^{14} + \frac{1}{6272} a^{12} + \frac{11}{896} a^{10} - \frac{187}{6272} a^{8} - \frac{285}{6272} a^{6} + \frac{85}{896} a^{4} - \frac{433}{6272} a^{2} - \frac{1049}{6272}$, $\frac{1}{12544} a^{15} - \frac{1}{12544} a^{14} + \frac{1}{12544} a^{13} - \frac{1}{12544} a^{12} + \frac{11}{1792} a^{11} - \frac{11}{1792} a^{10} + \frac{205}{12544} a^{9} - \frac{205}{12544} a^{8} + \frac{499}{12544} a^{7} - \frac{499}{12544} a^{6} + \frac{85}{1792} a^{5} - \frac{85}{1792} a^{4} - \frac{1217}{12544} a^{3} + \frac{1217}{12544} a^{2} + \frac{4831}{12544} a - \frac{4831}{12544}$, $\frac{1}{87808} a^{16} + \frac{1}{43904} a^{14} - \frac{3}{43904} a^{12} + \frac{337}{43904} a^{10} - \frac{559}{21952} a^{8} + \frac{2619}{43904} a^{6} - \frac{5309}{43904} a^{4} - \frac{5837}{43904} a^{2} + \frac{32187}{87808}$, $\frac{1}{175616} a^{17} - \frac{1}{175616} a^{16} + \frac{1}{87808} a^{15} - \frac{1}{87808} a^{14} - \frac{3}{87808} a^{13} + \frac{3}{87808} a^{12} + \frac{337}{87808} a^{11} - \frac{337}{87808} a^{10} + \frac{813}{43904} a^{9} - \frac{813}{43904} a^{8} - \frac{2869}{87808} a^{7} + \frac{2869}{87808} a^{6} - \frac{5309}{87808} a^{5} + \frac{5309}{87808} a^{4} + \frac{21603}{87808} a^{3} - \frac{21603}{87808} a^{2} + \frac{70603}{175616} a - \frac{70603}{175616}$, $\frac{1}{3744353114787653002584510432944640512} a^{18} + \frac{15327162185730933951279394533475}{3744353114787653002584510432944640512} a^{16} - \frac{923790544970687125559419239461}{117011034837114156330765951029520016} a^{14} + \frac{78732295087617400146575253253715}{468044139348456625323063804118080064} a^{12} - \frac{28206406836871654534834405898969435}{1872176557393826501292255216472320256} a^{10} - \frac{41197524576407801464720831376116869}{1872176557393826501292255216472320256} a^{8} + \frac{22607100770164777810206907828899833}{468044139348456625323063804118080064} a^{6} + \frac{6585952132157367252767038574809231}{58505517418557078165382975514760008} a^{4} + \frac{491780419263504553740175956773027725}{3744353114787653002584510432944640512} a^{2} + \frac{707645951337070555331525991980115535}{3744353114787653002584510432944640512}$, $\frac{1}{120011766912011211022343182859502716795597824} a^{19} - \frac{1}{7488706229575306005169020865889281024} a^{18} - \frac{230531338854439973564258118144340648485}{120011766912011211022343182859502716795597824} a^{17} - \frac{15327162185730933951279394533475}{7488706229575306005169020865889281024} a^{16} - \frac{149433768398573395176844046558072236321}{4286134532571828965083685102125097028414208} a^{15} - \frac{141858444565125008519460808111761}{1872176557393826501292255216472320256} a^{14} + \frac{12732039832343477528554781986598887848035}{30002941728002802755585795714875679198899456} a^{13} - \frac{306713359100125305817086668534879}{1872176557393826501292255216472320256} a^{12} - \frac{256754000184265190995612434234136068622297}{60005883456005605511171591429751358397798912} a^{11} - \frac{53283420996118561481234738568017719}{3744353114787653002584510432944640512} a^{10} - \frac{258169025459006290174891818893836423360901}{8572269065143657930167370204250194056828416} a^{9} + \frac{38511046735759772365289980459622787}{3744353114787653002584510432944640512} a^{8} - \frac{14138453414479943142976006057036220265571}{612304933224546995011955014589299575487744} a^{7} + \frac{55827215021821316619290966034783307}{1872176557393826501292255216472320256} a^{6} - \frac{1052516304533539290195335531798207682958881}{30002941728002802755585795714875679198899456} a^{5} - \frac{135672734206270648665631658088519843}{1872176557393826501292255216472320256} a^{4} - \frac{2719865887614897408026647504602003039077665}{17144538130287315860334740408500388113656832} a^{3} + \frac{1521883971071118146788770741301314183}{7488706229575306005169020865889281024} a^{2} - \frac{13788358611987419585285240684237899208678557}{120011766912011211022343182859502716795597824} a + \frac{1673767405628482350808399409329860709}{7488706229575306005169020865889281024}$

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

Class group and class number

$C_{2}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{4}\times C_{434620}$, which has order $890101760$ (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:  \( 19344397.966990974 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-210}) \), \(\Q(\sqrt{-21}) \), \(\Q(\sqrt{10}) \), \(\Q(\sqrt{10}, \sqrt{-21})\), 5.5.390625.1, 10.0.102102525000000000000000.1, 10.0.638140781250000000000.3, 10.10.25000000000000000.1

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

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 ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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}$
$5$5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$