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Properties

Label 20.0.59699157574...0000.5
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 5^{10}\cdot 11^{18}$
Root discriminant $77.41$
Ramified primes $2, 5, 11$
Class number $22692$ (GRH)
Class group $[2, 11346]$ (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![7144929, 0, 15877620, 0, 18612099, 0, 10624284, 0, 3758139, 0, 855536, 0, 132880, 0, 14036, 0, 1001, 0, 44, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 44*x^18 + 1001*x^16 + 14036*x^14 + 132880*x^12 + 855536*x^10 + 3758139*x^8 + 10624284*x^6 + 18612099*x^4 + 15877620*x^2 + 7144929)
 
gp: K = bnfinit(x^20 + 44*x^18 + 1001*x^16 + 14036*x^14 + 132880*x^12 + 855536*x^10 + 3758139*x^8 + 10624284*x^6 + 18612099*x^4 + 15877620*x^2 + 7144929, 1)
 

Normalized defining polynomial

\( x^{20} + 44 x^{18} + 1001 x^{16} + 14036 x^{14} + 132880 x^{12} + 855536 x^{10} + 3758139 x^{8} + 10624284 x^{6} + 18612099 x^{4} + 15877620 x^{2} + 7144929 \)

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:  \(59699157574783284402391613440000000000=2^{40}\cdot 5^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $77.41$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(440=2^{3}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{440}(1,·)$, $\chi_{440}(131,·)$, $\chi_{440}(199,·)$, $\chi_{440}(201,·)$, $\chi_{440}(371,·)$, $\chi_{440}(399,·)$, $\chi_{440}(81,·)$, $\chi_{440}(211,·)$, $\chi_{440}(149,·)$, $\chi_{440}(279,·)$, $\chi_{440}(349,·)$, $\chi_{440}(159,·)$, $\chi_{440}(401,·)$, $\chi_{440}(361,·)$, $\chi_{440}(171,·)$, $\chi_{440}(109,·)$, $\chi_{440}(29,·)$, $\chi_{440}(51,·)$, $\chi_{440}(119,·)$, $\chi_{440}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10}$, $\frac{1}{33} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{99} a^{12} - \frac{1}{99} a^{10} + \frac{1}{9} a^{8} - \frac{2}{9} a^{6} + \frac{2}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{297} a^{13} - \frac{1}{297} a^{11} - \frac{8}{27} a^{9} - \frac{11}{27} a^{7} - \frac{7}{27} a^{5} + \frac{7}{27} a^{3}$, $\frac{1}{20493} a^{14} - \frac{19}{20493} a^{12} + \frac{821}{20493} a^{10} - \frac{245}{1863} a^{8} - \frac{916}{1863} a^{6} + \frac{295}{1863} a^{4} + \frac{76}{207} a^{2} + \frac{5}{23}$, $\frac{1}{61479} a^{15} - \frac{19}{61479} a^{13} + \frac{821}{61479} a^{11} - \frac{2108}{5589} a^{9} - \frac{2779}{5589} a^{7} + \frac{2158}{5589} a^{5} - \frac{131}{621} a^{3} + \frac{5}{69} a$, $\frac{1}{184437} a^{16} - \frac{1}{184437} a^{14} + \frac{479}{184437} a^{12} + \frac{8357}{184437} a^{10} - \frac{1600}{16767} a^{8} + \frac{8026}{16767} a^{6} + \frac{17}{69} a^{4} - \frac{50}{207} a^{2} + \frac{10}{23}$, $\frac{1}{553311} a^{17} - \frac{1}{553311} a^{15} + \frac{479}{553311} a^{13} + \frac{8357}{553311} a^{11} - \frac{18367}{50301} a^{9} + \frac{24793}{50301} a^{7} + \frac{17}{207} a^{5} + \frac{157}{621} a^{3} + \frac{11}{23} a$, $\frac{1}{968833717219090899} a^{18} + \frac{1296580422851}{968833717219090899} a^{16} - \frac{7945269691948}{968833717219090899} a^{14} + \frac{1689815945459354}{968833717219090899} a^{12} + \frac{41194701074032159}{968833717219090899} a^{10} - \frac{27479567702638757}{88075792474462809} a^{8} - \frac{1102809085637368}{9786199163829201} a^{6} - \frac{386984218285447}{1087355462647689} a^{4} + \frac{18052396639025}{120817273627521} a^{2} - \frac{1904214646}{13424141514169}$, $\frac{1}{2906501151657272697} a^{19} + \frac{1296580422851}{2906501151657272697} a^{17} - \frac{7945269691948}{2906501151657272697} a^{15} + \frac{1689815945459354}{2906501151657272697} a^{13} + \frac{41194701074032159}{2906501151657272697} a^{11} - \frac{27479567702638757}{264227377423388427} a^{9} - \frac{1102809085637368}{29358597491487603} a^{7} - \frac{1474339680933136}{3262066387943067} a^{5} - \frac{102764876988496}{362451820882563} a^{3} + \frac{4474079099841}{13424141514169} a$

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

Class group and class number

$C_{2}\times C_{11346}$, which has order $22692$ (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:  \( 1589230.00872 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{22}) \), \(\Q(\sqrt{-110}) \), \(\Q(\sqrt{-5}, \sqrt{22})\), \(\Q(\zeta_{11})^+\), 10.0.685948419200000.1, 10.10.77265229938688.1, 10.0.241453843558400000.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
11Data not computed