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Properties

Label 20.14.5140615083...6064.1
Degree $20$
Signature $[14, 3]$
Discriminant $-\,2^{46}\cdot 31^{7}\cdot 227^{4}$
Root discriminant $48.48$
Ramified primes $2, 31, 227$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1037

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9, -456, 5824, -20996, 16936, 34260, -54194, -12160, 48973, -3352, -20502, 552, 5001, 1492, -1036, -604, 194, 84, -22, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 22*x^18 + 84*x^17 + 194*x^16 - 604*x^15 - 1036*x^14 + 1492*x^13 + 5001*x^12 + 552*x^11 - 20502*x^10 - 3352*x^9 + 48973*x^8 - 12160*x^7 - 54194*x^6 + 34260*x^5 + 16936*x^4 - 20996*x^3 + 5824*x^2 - 456*x - 9)
 
gp: K = bnfinit(x^20 - 4*x^19 - 22*x^18 + 84*x^17 + 194*x^16 - 604*x^15 - 1036*x^14 + 1492*x^13 + 5001*x^12 + 552*x^11 - 20502*x^10 - 3352*x^9 + 48973*x^8 - 12160*x^7 - 54194*x^6 + 34260*x^5 + 16936*x^4 - 20996*x^3 + 5824*x^2 - 456*x - 9, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 22 x^{18} + 84 x^{17} + 194 x^{16} - 604 x^{15} - 1036 x^{14} + 1492 x^{13} + 5001 x^{12} + 552 x^{11} - 20502 x^{10} - 3352 x^{9} + 48973 x^{8} - 12160 x^{7} - 54194 x^{6} + 34260 x^{5} + 16936 x^{4} - 20996 x^{3} + 5824 x^{2} - 456 x - 9 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-5140615083075201996699936766296064=-\,2^{46}\cdot 31^{7}\cdot 227^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.48$
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:  $2, 31, 227$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{16} - \frac{1}{3} a^{15} + \frac{1}{3} a^{14} - \frac{1}{3} a^{12} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{1781439322156942087015223030201505} a^{19} - \frac{18515177772135182272581321375518}{118762621477129472467681535346767} a^{18} + \frac{264915284899439644231739093510003}{1781439322156942087015223030201505} a^{17} - \frac{169026001476963562929792672912739}{1781439322156942087015223030201505} a^{16} + \frac{552426860756547263034120809276818}{1781439322156942087015223030201505} a^{15} - \frac{620060424116658683488215576629}{2365789272452778336009592337585} a^{14} + \frac{283909565085215847175162672102676}{1781439322156942087015223030201505} a^{13} - \frac{34676882741221450339313834461778}{593813107385647362338407676733835} a^{12} - \frac{22077927759087483367460743486489}{118762621477129472467681535346767} a^{11} - \frac{101712851538219432918759082587106}{593813107385647362338407676733835} a^{10} + \frac{133993376495899100236722493100632}{593813107385647362338407676733835} a^{9} - \frac{27045060130156099748766063220153}{1781439322156942087015223030201505} a^{8} + \frac{26326982109874303059870587939877}{593813107385647362338407676733835} a^{7} + \frac{135306128599736805918862910001839}{1781439322156942087015223030201505} a^{6} - \frac{24755533943110059982988431885626}{593813107385647362338407676733835} a^{5} - \frac{120407277577465155845878521526724}{593813107385647362338407676733835} a^{4} - \frac{316905355394235100355800973767817}{1781439322156942087015223030201505} a^{3} + \frac{163174677846617928425412244440416}{1781439322156942087015223030201505} a^{2} - \frac{214446103746206579215641692454354}{593813107385647362338407676733835} a - \frac{224835584391129803912757434821248}{593813107385647362338407676733835}$

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:  $16$
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:  \( 20295301629.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1037:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 14745600
The 384 conjugacy class representatives for t20n1037 are not computed
Character table for t20n1037 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.10.207699287474176.1

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 $16{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}$ $16{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.10.0.1}{10} }{,}\,{\href{/LocalNumberField/23.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ R ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$2$2.8.22.7$x^{8} + 2 x^{4} + 16 x + 4$$4$$2$$22$$C_4\times C_2$$[3, 4]^{2}$
2.12.24.342$x^{12} + 4 x^{11} + 2 x^{10} + 4 x^{9} + 4 x^{5} - 2 x^{4} + 4 x^{3} - 2 x^{2} + 4 x - 2$$12$$1$$24$$C_2 \times S_4$$[4/3, 4/3, 3]_{3}^{2}$
31Data not computed
227Data not computed