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Properties

Label 20.0.10473339213...5568.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{27}\cdot 727^{8}$
Root discriminant $35.56$
Ramified primes $2, 727$
Class number $11$ (GRH)
Class group $[11]$ (GRH)
Galois group 20T525

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51200, 0, 16896, 0, 14528, 0, -64, 0, 1092, 0, 72, 0, 265, 0, 4, 0, -6, 0, -4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^18 - 6*x^16 + 4*x^14 + 265*x^12 + 72*x^10 + 1092*x^8 - 64*x^6 + 14528*x^4 + 16896*x^2 + 51200)
 
gp: K = bnfinit(x^20 - 4*x^18 - 6*x^16 + 4*x^14 + 265*x^12 + 72*x^10 + 1092*x^8 - 64*x^6 + 14528*x^4 + 16896*x^2 + 51200, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{18} - 6 x^{16} + 4 x^{14} + 265 x^{12} + 72 x^{10} + 1092 x^{8} - 64 x^{6} + 14528 x^{4} + 16896 x^{2} + 51200 \)

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:  \(10473339213126553396125334765568=2^{27}\cdot 727^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.56$
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:  $2, 727$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{10} + \frac{1}{16} a^{6} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{11} - \frac{1}{16} a^{9} - \frac{3}{32} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{12} + \frac{1}{32} a^{8} - \frac{1}{8} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{9} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} - \frac{1}{8} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{64} a^{14} - \frac{1}{64} a^{12} - \frac{1}{64} a^{10} + \frac{1}{64} a^{8} - \frac{1}{16} a^{6} + \frac{3}{16} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{15} - \frac{1}{128} a^{14} + \frac{3}{256} a^{13} + \frac{1}{128} a^{12} - \frac{1}{256} a^{11} + \frac{1}{128} a^{10} - \frac{3}{256} a^{9} + \frac{7}{128} a^{8} + \frac{5}{64} a^{7} - \frac{3}{32} a^{6} + \frac{5}{64} a^{5} - \frac{5}{32} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{256} a^{16} - \frac{1}{256} a^{14} + \frac{3}{256} a^{12} + \frac{1}{256} a^{10} - \frac{1}{16} a^{8} - \frac{1}{8} a^{7} + \frac{1}{64} a^{6} - \frac{1}{8} a^{5} - \frac{3}{16} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{256} a^{17} - \frac{1}{128} a^{14} - \frac{1}{128} a^{13} + \frac{1}{128} a^{12} + \frac{1}{128} a^{10} + \frac{5}{256} a^{9} + \frac{7}{128} a^{8} - \frac{3}{32} a^{7} - \frac{3}{32} a^{6} - \frac{7}{64} a^{5} - \frac{5}{32} a^{4}$, $\frac{1}{872438346752} a^{18} - \frac{162322207}{109054793344} a^{16} - \frac{3126267183}{436219173376} a^{14} - \frac{3102283139}{218109586688} a^{12} + \frac{4933726225}{872438346752} a^{10} - \frac{8927574913}{218109586688} a^{8} - \frac{1}{8} a^{7} + \frac{15726773445}{218109586688} a^{6} - \frac{1}{8} a^{5} + \frac{247285483}{7789628096} a^{4} - \frac{909408109}{3407962292} a^{2} - \frac{593357399}{1703981146}$, $\frac{1}{17448766935040} a^{19} - \frac{1}{1744876693504} a^{18} + \frac{299206189}{272636983360} a^{17} + \frac{162322207}{218109586688} a^{16} + \frac{13913544277}{8724383467520} a^{15} + \frac{3126267183}{872438346752} a^{14} + \frac{5417622591}{4362191733760} a^{13} - \frac{3713641445}{436219173376} a^{12} - \frac{5829179339}{3489753387008} a^{11} - \frac{4933726225}{1744876693504} a^{10} + \frac{120574992183}{4362191733760} a^{9} - \frac{25152048007}{436219173376} a^{8} + \frac{131597491373}{4362191733760} a^{7} + \frac{52432472395}{436219173376} a^{6} + \frac{14852838163}{155792561920} a^{5} + \frac{3647528565}{15579256192} a^{4} + \frac{19538365643}{68159245840} a^{3} - \frac{2498554183}{6815924584} a^{2} - \frac{5705300837}{34079622920} a - \frac{1110623747}{3407962292}$

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

Class group and class number

$C_{11}$, which has order $11$ (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:  \( -\frac{19885049}{872438346752} a^{18} + \frac{10380037}{109054793344} a^{16} + \frac{31456503}{436219173376} a^{14} + \frac{16289995}{218109586688} a^{12} - \frac{5295150793}{872438346752} a^{10} + \frac{71783421}{218109586688} a^{8} - \frac{8603999197}{218109586688} a^{6} + \frac{93399585}{7789628096} a^{4} - \frac{512514509}{851990573} a^{2} - \frac{446903797}{1703981146} \) (order $4$)
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:  \( 67262537.361 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T525:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20480
The 152 conjugacy class representatives for t20n525 are not computed
Character table for t20n525 is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), 5.5.8456464.1, 10.0.286047133533184.1

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 $20$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{6}$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ $20$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.4.6.8$x^{4} + 2 x^{3} + 2$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.4.9.4$x^{4} + 2 x^{2} + 10$$4$$1$$9$$D_{4}$$[2, 3, 7/2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
727Data not computed