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Properties

Label 20.0.16863534401...000.10
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $204.81$
Ramified primes $2, 5, 7, 11$
Class number $273493616$ (GRH)
Class group $[2, 2, 2, 34186702]$ (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![21907295553529, 0, 8639912141640, 0, 1526410518469, 0, 158822764880, 0, 10777388986, 0, 501223744, 0, 16551690, 0, 407472, 0, 7965, 0, 120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 120*x^18 + 7965*x^16 + 407472*x^14 + 16551690*x^12 + 501223744*x^10 + 10777388986*x^8 + 158822764880*x^6 + 1526410518469*x^4 + 8639912141640*x^2 + 21907295553529)
 
gp: K = bnfinit(x^20 + 120*x^18 + 7965*x^16 + 407472*x^14 + 16551690*x^12 + 501223744*x^10 + 10777388986*x^8 + 158822764880*x^6 + 1526410518469*x^4 + 8639912141640*x^2 + 21907295553529, 1)
 

Normalized defining polynomial

\( x^{20} + 120 x^{18} + 7965 x^{16} + 407472 x^{14} + 16551690 x^{12} + 501223744 x^{10} + 10777388986 x^{8} + 158822764880 x^{6} + 1526410518469 x^{4} + 8639912141640 x^{2} + 21907295553529 \)

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:  \(16863534401027144382603387201975746560000000000=2^{40}\cdot 5^{10}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $204.81$
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:  $2, 5, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3080=2^{3}\cdot 5\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3080}(1,·)$, $\chi_{3080}(3011,·)$, $\chi_{3080}(69,·)$, $\chi_{3080}(3079,·)$, $\chi_{3080}(841,·)$, $\chi_{3080}(1679,·)$, $\chi_{3080}(1681,·)$, $\chi_{3080}(211,·)$, $\chi_{3080}(1051,·)$, $\chi_{3080}(2589,·)$, $\chi_{3080}(1119,·)$, $\chi_{3080}(1891,·)$, $\chi_{3080}(1189,·)$, $\chi_{3080}(1961,·)$, $\chi_{3080}(491,·)$, $\chi_{3080}(2029,·)$, $\chi_{3080}(2869,·)$, $\chi_{3080}(1399,·)$, $\chi_{3080}(1401,·)$, $\chi_{3080}(2239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} + \frac{1}{4}$, $\frac{1}{4} a^{5} + \frac{1}{4} a$, $\frac{1}{4} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} - \frac{3}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} - \frac{3}{16} a$, $\frac{1}{16} a^{10} - \frac{1}{8} a^{6} - \frac{3}{16} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{8} a^{7} - \frac{3}{16} a^{3}$, $\frac{1}{64} a^{12} - \frac{1}{64} a^{8} - \frac{5}{64} a^{4} - \frac{3}{64}$, $\frac{1}{64} a^{13} - \frac{1}{64} a^{9} - \frac{5}{64} a^{5} - \frac{3}{64} a$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{10} - \frac{5}{64} a^{6} - \frac{3}{64} a^{2}$, $\frac{1}{1472} a^{15} - \frac{1}{184} a^{13} + \frac{7}{1472} a^{11} + \frac{1}{46} a^{9} + \frac{91}{1472} a^{7} - \frac{7}{184} a^{5} - \frac{13}{64} a^{3} - \frac{11}{46} a$, $\frac{1}{5888} a^{16} - \frac{1}{736} a^{14} - \frac{1}{368} a^{12} - \frac{19}{736} a^{10} - \frac{35}{2944} a^{8} + \frac{85}{736} a^{6} + \frac{3}{32} a^{4} + \frac{71}{736} a^{2} - \frac{85}{256}$, $\frac{1}{5888} a^{17} + \frac{3}{1472} a^{13} - \frac{3}{184} a^{11} + \frac{47}{2944} a^{9} - \frac{1}{92} a^{7} - \frac{89}{1472} a^{5} + \frac{81}{184} a^{3} + \frac{841}{5888} a$, $\frac{1}{1243239642240600050737522991121152} a^{18} + \frac{103077345856185080710644204563}{1243239642240600050737522991121152} a^{16} - \frac{65217024551177509784402087833}{9712809705004687896386898368134} a^{14} - \frac{73409153552432936057494288611}{310809910560150012684380747780288} a^{12} + \frac{3077891663987176427682188459165}{621619821120300025368761495560576} a^{10} + \frac{7409373864232348879612641783781}{621619821120300025368761495560576} a^{8} - \frac{10115045460695306508440885726793}{155404955280075006342190373890144} a^{6} + \frac{3338512204322820558075485262433}{310809910560150012684380747780288} a^{4} - \frac{3908375579835657359775606041603}{1243239642240600050737522991121152} a^{2} - \frac{25169177001605454147866416056675}{54053897488721741336414043092224}$, $\frac{1}{253000510435604350925136666216145553152} a^{19} + \frac{522841817946595780035980324477653}{31625063804450543865642083277018194144} a^{17} + \frac{775447300018639015122199815968911}{15812531902225271932821041638509097072} a^{15} + \frac{16617292243241674972785905264680405}{2750005548213090770925398545827669056} a^{13} - \frac{1984638133119946945176933561046790587}{126500255217802175462568333108072776576} a^{11} + \frac{40036110217764663504282107409039939}{2750005548213090770925398545827669056} a^{9} + \frac{2587396664264739640654310565238704233}{31625063804450543865642083277018194144} a^{7} - \frac{2718828313508562706343480433927813063}{63250127608901087731284166554036388288} a^{5} - \frac{84560248847278314653470893479623101523}{253000510435604350925136666216145553152} a^{3} + \frac{24334985224185231168286424960581709037}{63250127608901087731284166554036388288} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{34186702}$, which has order $273493616$ (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:  \( 1589230.0087159988 \) (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

\(\Q(\sqrt{-385}) \), \(\Q(\sqrt{-70}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{22}, \sqrt{-70})\), \(\Q(\zeta_{11})^+\), 10.0.126816085896438400000.1, 10.0.368919522607820800000.1, 10.10.77265229938688.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
5Data not computed
$7$7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
7.10.5.1$x^{10} - 98 x^{6} + 2401 x^{2} - 268912$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed