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Properties

Label 20.0.55972477611...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 211^{10}$
Root discriminant $97.14$
Ramified primes $2, 5, 211$
Class number $576632$ (GRH)
Class group $[2, 2, 144158]$ (GRH)
Galois group 20T534

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![139128125, 0, 485959375, 0, 639989375, 0, 409603750, 0, 140420500, 0, 26876125, 0, 2949850, 0, 188100, 0, 6830, 0, 130, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 130*x^18 + 6830*x^16 + 188100*x^14 + 2949850*x^12 + 26876125*x^10 + 140420500*x^8 + 409603750*x^6 + 639989375*x^4 + 485959375*x^2 + 139128125)
 
gp: K = bnfinit(x^20 + 130*x^18 + 6830*x^16 + 188100*x^14 + 2949850*x^12 + 26876125*x^10 + 140420500*x^8 + 409603750*x^6 + 639989375*x^4 + 485959375*x^2 + 139128125, 1)
 

Normalized defining polynomial

\( x^{20} + 130 x^{18} + 6830 x^{16} + 188100 x^{14} + 2949850 x^{12} + 26876125 x^{10} + 140420500 x^{8} + 409603750 x^{6} + 639989375 x^{4} + 485959375 x^{2} + 139128125 \)

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:  \(5597247761133055315411232000000000000000=2^{20}\cdot 5^{15}\cdot 211^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97.14$
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:  $2, 5, 211$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{88871843693027150839136875} a^{18} - \frac{22636186839207420525176}{88871843693027150839136875} a^{16} - \frac{55971505273784345900456}{17774368738605430167827375} a^{14} + \frac{58440995298232666520508}{17774368738605430167827375} a^{12} + \frac{6912225776450160393167}{710974749544217206713095} a^{10} + \frac{203080619540023843657}{16847742880194720538225} a^{8} + \frac{200475712718105301838}{3369548576038944107645} a^{6} + \frac{130381430614071606508}{3369548576038944107645} a^{4} - \frac{84335246060080452814}{673909715207788821529} a^{2} + \frac{122071780925943364375}{673909715207788821529}$, $\frac{1}{88871843693027150839136875} a^{19} - \frac{22636186839207420525176}{88871843693027150839136875} a^{17} - \frac{55971505273784345900456}{17774368738605430167827375} a^{15} + \frac{58440995298232666520508}{17774368738605430167827375} a^{13} + \frac{6912225776450160393167}{710974749544217206713095} a^{11} + \frac{203080619540023843657}{16847742880194720538225} a^{9} + \frac{200475712718105301838}{3369548576038944107645} a^{7} + \frac{130381430614071606508}{3369548576038944107645} a^{5} - \frac{84335246060080452814}{673909715207788821529} a^{3} + \frac{122071780925943364375}{673909715207788821529} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{144158}$, which has order $576632$ (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:  \( 454775.127492 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T534:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 25600
The 88 conjugacy class representatives for t20n534 are not computed
Character table for t20n534 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.6194123253125.1

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 $20$ R $20$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
211Data not computed