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Properties

Label 20.0.28352955239...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{14}\cdot 11^{11}\cdot 353^{4}$
Root discriminant $74.58$
Ramified primes $2, 5, 11, 353$
Class number $13072$ (GRH)
Class group $[2, 6536]$ (GRH)
Galois group 20T658

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1331, 0, 17061, 0, 83413, 0, 200233, 0, 255321, 0, 175867, 0, 64301, 0, 12323, 0, 1198, 0, 56, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 56*x^18 + 1198*x^16 + 12323*x^14 + 64301*x^12 + 175867*x^10 + 255321*x^8 + 200233*x^6 + 83413*x^4 + 17061*x^2 + 1331)
 
gp: K = bnfinit(x^20 + 56*x^18 + 1198*x^16 + 12323*x^14 + 64301*x^12 + 175867*x^10 + 255321*x^8 + 200233*x^6 + 83413*x^4 + 17061*x^2 + 1331, 1)
 

Normalized defining polynomial

\( x^{20} + 56 x^{18} + 1198 x^{16} + 12323 x^{14} + 64301 x^{12} + 175867 x^{10} + 255321 x^{8} + 200233 x^{6} + 83413 x^{4} + 17061 x^{2} + 1331 \)

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:  \(28352955239860252353862400000000000000=2^{20}\cdot 5^{14}\cdot 11^{11}\cdot 353^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.58$
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:  $2, 5, 11, 353$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{2}{5} a^{10} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{11} - \frac{1}{5} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} + \frac{1}{5} a$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{8} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4} + \frac{2}{5} a^{2} - \frac{2}{5}$, $\frac{1}{5} a^{15} - \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{55} a^{16} + \frac{1}{55} a^{14} - \frac{1}{55} a^{12} - \frac{19}{55} a^{10} + \frac{6}{55} a^{8} + \frac{2}{11} a^{6} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{55} a^{17} + \frac{1}{55} a^{15} - \frac{1}{55} a^{13} - \frac{19}{55} a^{11} + \frac{6}{55} a^{9} + \frac{2}{11} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{1640373174748495} a^{18} + \frac{7103563230479}{1640373174748495} a^{16} - \frac{24242570009384}{1640373174748495} a^{14} + \frac{32637290962523}{328074634949699} a^{12} - \frac{53024867579722}{1640373174748495} a^{10} + \frac{452824731247503}{1640373174748495} a^{8} - \frac{65565653005728}{149124834068045} a^{6} + \frac{61919881980383}{149124834068045} a^{4} + \frac{24875784683963}{149124834068045} a^{2} + \frac{715896226501}{2711360619419}$, $\frac{1}{1640373174748495} a^{19} + \frac{7103563230479}{1640373174748495} a^{17} - \frac{24242570009384}{1640373174748495} a^{15} + \frac{32637290962523}{328074634949699} a^{13} - \frac{53024867579722}{1640373174748495} a^{11} + \frac{452824731247503}{1640373174748495} a^{9} - \frac{65565653005728}{149124834068045} a^{7} + \frac{61919881980383}{149124834068045} a^{5} + \frac{24875784683963}{149124834068045} a^{3} + \frac{715896226501}{2711360619419} a$

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

Class group and class number

$C_{2}\times C_{6536}$, which has order $13072$ (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:  \( 3436243.55903 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T658:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 76 conjugacy class representatives for t20n658 are not computed
Character table for t20n658 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.4400.1, 10.10.142531278828125.1

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 24 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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.10.0.1}{10} }{,}\,{\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.6.5.1$x^{6} - 11$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
353Data not computed