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Properties

Label 20.0.26233125242...2352.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{15}\cdot 11491^{4}$
Root discriminant $29.58$
Ramified primes $2, 3, 11491$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T466

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, 162, 0, 1458, 0, -2232, 0, 2277, 0, -2073, 0, 1318, 0, -524, 0, 126, 0, -17, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 17*x^18 + 126*x^16 - 524*x^14 + 1318*x^12 - 2073*x^10 + 2277*x^8 - 2232*x^6 + 1458*x^4 + 162*x^2 + 27)
 
gp: K = bnfinit(x^20 - 17*x^18 + 126*x^16 - 524*x^14 + 1318*x^12 - 2073*x^10 + 2277*x^8 - 2232*x^6 + 1458*x^4 + 162*x^2 + 27, 1)
 

Normalized defining polynomial

\( x^{20} - 17 x^{18} + 126 x^{16} - 524 x^{14} + 1318 x^{12} - 2073 x^{10} + 2277 x^{8} - 2232 x^{6} + 1458 x^{4} + 162 x^{2} + 27 \)

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:  \(262331252425064668558343012352=2^{20}\cdot 3^{15}\cdot 11491^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.58$
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:  $2, 3, 11491$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{7} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{12} + \frac{1}{9} a^{8} + \frac{4}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} + \frac{1}{9} a^{9} + \frac{4}{9} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{27} a^{16} + \frac{1}{27} a^{14} - \frac{1}{9} a^{12} + \frac{4}{27} a^{10} + \frac{4}{27} a^{8} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} + \frac{1}{3}$, $\frac{1}{81} a^{17} - \frac{2}{81} a^{15} - \frac{2}{27} a^{13} - \frac{5}{81} a^{11} - \frac{8}{81} a^{9} + \frac{2}{9} a^{7} - \frac{13}{27} a^{5} - \frac{4}{9} a^{3} + \frac{4}{9} a$, $\frac{1}{3352509} a^{18} + \frac{55822}{3352509} a^{16} - \frac{2071}{41389} a^{14} - \frac{262634}{3352509} a^{12} + \frac{518263}{3352509} a^{10} - \frac{147091}{1117503} a^{8} + \frac{67904}{1117503} a^{6} + \frac{337}{124167} a^{4} - \frac{1205}{372501} a^{2} - \frac{37210}{124167}$, $\frac{1}{3352509} a^{19} + \frac{4811}{1117503} a^{17} - \frac{84973}{3352509} a^{15} - \frac{14300}{3352509} a^{13} - \frac{130765}{1117503} a^{11} - \frac{110161}{3352509} a^{9} - \frac{180430}{1117503} a^{7} + \frac{168589}{1117503} a^{5} + \frac{164351}{372501} a^{3} + \frac{95315}{372501} a$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( -\frac{6934}{1117503} a^{18} + \frac{82958}{1117503} a^{16} - \frac{127841}{372501} a^{14} + \frac{691769}{1117503} a^{12} + \frac{378173}{1117503} a^{10} - \frac{1054858}{372501} a^{8} + \frac{1360765}{372501} a^{6} - \frac{184530}{41389} a^{4} + \frac{905450}{124167} a^{2} + \frac{36503}{41389} \) (order $6$)
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:  \( 34471446.634 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T466:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15360
The 90 conjugacy class representatives for t20n466 are not computed
Character table for t20n466 is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 5.5.310257.1, 10.0.288778218147.1

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 R ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.4.3.1$x^{4} + 3$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
3.8.6.2$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$$4$$2$$6$$D_4$$[\ ]_{4}^{2}$
11491Data not computed