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Properties

Label 20.10.3893333459...1984.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{4}\cdot 1093^{10}$
Root discriminant $37.98$
Ramified primes $2, 1093$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T423

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 64, 0, -191, 0, -292, 0, 1033, 0, 38, 0, -685, 0, -10, 0, 89, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 18*x^18 + 89*x^16 - 10*x^14 - 685*x^12 + 38*x^10 + 1033*x^8 - 292*x^6 - 191*x^4 + 64*x^2 - 1)
 
gp: K = bnfinit(x^20 + 18*x^18 + 89*x^16 - 10*x^14 - 685*x^12 + 38*x^10 + 1033*x^8 - 292*x^6 - 191*x^4 + 64*x^2 - 1, 1)
 

Normalized defining polynomial

\( x^{20} + 18 x^{18} + 89 x^{16} - 10 x^{14} - 685 x^{12} + 38 x^{10} + 1033 x^{8} - 292 x^{6} - 191 x^{4} + 64 x^{2} - 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-38933334597022895851373555971984=-\,2^{4}\cdot 1093^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $37.98$
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:  $2, 1093$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{6} + \frac{3}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a + \frac{3}{8}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{7} - \frac{1}{4} a^{6} - \frac{1}{8} a^{5} - \frac{1}{2} a^{4} + \frac{1}{8} a^{3} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{8} a^{12} - \frac{1}{4} a^{7} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{1}{16} a^{5} - \frac{7}{16} a^{4} + \frac{7}{16} a^{3} + \frac{1}{16} a^{2} + \frac{3}{8} a - \frac{5}{16}$, $\frac{1}{80} a^{14} - \frac{3}{80} a^{12} - \frac{1}{16} a^{11} - \frac{1}{80} a^{10} - \frac{1}{16} a^{9} + \frac{3}{80} a^{8} - \frac{3}{16} a^{7} + \frac{11}{80} a^{6} + \frac{1}{16} a^{5} - \frac{3}{16} a^{4} - \frac{7}{16} a^{3} - \frac{9}{20} a^{2} + \frac{3}{16} a + \frac{2}{5}$, $\frac{1}{80} a^{15} + \frac{1}{40} a^{13} - \frac{1}{16} a^{12} + \frac{1}{20} a^{11} - \frac{1}{40} a^{9} - \frac{1}{8} a^{8} + \frac{3}{40} a^{7} - \frac{1}{4} a^{5} - \frac{1}{2} a^{4} + \frac{9}{80} a^{3} + \frac{1}{8} a^{2} + \frac{2}{5} a - \frac{7}{16}$, $\frac{1}{640} a^{16} + \frac{1}{640} a^{14} + \frac{27}{640} a^{12} - \frac{1}{16} a^{11} - \frac{13}{320} a^{10} - \frac{1}{16} a^{9} - \frac{1}{320} a^{8} - \frac{3}{16} a^{7} + \frac{17}{320} a^{6} + \frac{1}{16} a^{5} + \frac{89}{640} a^{4} - \frac{7}{16} a^{3} - \frac{287}{640} a^{2} + \frac{3}{16} a + \frac{163}{640}$, $\frac{1}{640} a^{17} + \frac{1}{640} a^{15} - \frac{13}{640} a^{13} - \frac{1}{16} a^{12} + \frac{7}{320} a^{11} - \frac{21}{320} a^{9} - \frac{1}{8} a^{8} - \frac{3}{320} a^{7} - \frac{1}{4} a^{6} + \frac{49}{640} a^{5} + \frac{73}{640} a^{3} + \frac{1}{8} a^{2} - \frac{77}{640} a - \frac{3}{16}$, $\frac{1}{24320} a^{18} - \frac{1}{1280} a^{17} + \frac{1}{3040} a^{16} + \frac{7}{1280} a^{15} - \frac{43}{12160} a^{14} + \frac{29}{1280} a^{13} - \frac{917}{24320} a^{12} - \frac{31}{640} a^{11} + \frac{37}{3040} a^{10} + \frac{53}{640} a^{9} - \frac{15}{1216} a^{8} + \frac{67}{640} a^{7} + \frac{727}{24320} a^{6} - \frac{129}{1280} a^{5} + \frac{29}{80} a^{4} + \frac{559}{1280} a^{3} + \frac{1377}{12160} a^{2} - \frac{67}{1280} a - \frac{2339}{24320}$, $\frac{1}{24320} a^{19} - \frac{11}{24320} a^{17} - \frac{1}{1280} a^{16} + \frac{47}{24320} a^{15} + \frac{7}{1280} a^{14} - \frac{183}{12160} a^{13} - \frac{51}{1280} a^{12} + \frac{319}{12160} a^{11} - \frac{31}{640} a^{10} - \frac{1423}{12160} a^{9} - \frac{27}{640} a^{8} - \frac{4327}{24320} a^{7} + \frac{67}{640} a^{6} + \frac{51}{256} a^{5} + \frac{511}{1280} a^{4} + \frac{1155}{4864} a^{3} - \frac{561}{1280} a^{2} - \frac{2043}{6080} a - \frac{627}{1280}$

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:  $14$
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:  \( 951042096.622 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T423:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 160 conjugacy class representatives for t20n423 are not computed
Character table for t20n423 is not computed

Intermediate fields

\(\Q(\sqrt{1093}) \), 5.5.1194649.1 x5, 10.10.1559914552888693.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

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
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.0.1$x^{4} - x + 1$$1$$4$$0$$C_4$$[\ ]^{4}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
1093Data not computed