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Properties

Label 20.0.23108400064...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 27517559^{2}$
Root discriminant $18.54$
Ramified primes $5, 27517559$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T654

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, 8, -2, 19, -8, 52, -15, 109, -63, 196, -124, 157, -66, 67, -28, 29, -9, 7, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 7*x^18 - 9*x^17 + 29*x^16 - 28*x^15 + 67*x^14 - 66*x^13 + 157*x^12 - 124*x^11 + 196*x^10 - 63*x^9 + 109*x^8 - 15*x^7 + 52*x^6 - 8*x^5 + 19*x^4 - 2*x^3 + 8*x^2 + x + 1)
 
gp: K = bnfinit(x^20 - x^19 + 7*x^18 - 9*x^17 + 29*x^16 - 28*x^15 + 67*x^14 - 66*x^13 + 157*x^12 - 124*x^11 + 196*x^10 - 63*x^9 + 109*x^8 - 15*x^7 + 52*x^6 - 8*x^5 + 19*x^4 - 2*x^3 + 8*x^2 + x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 7 x^{18} - 9 x^{17} + 29 x^{16} - 28 x^{15} + 67 x^{14} - 66 x^{13} + 157 x^{12} - 124 x^{11} + 196 x^{10} - 63 x^{9} + 109 x^{8} - 15 x^{7} + 52 x^{6} - 8 x^{5} + 19 x^{4} - 2 x^{3} + 8 x^{2} + x + 1 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 10]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(23108400064650909423828125=5^{15}\cdot 27517559^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.54$
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:  $5, 27517559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{11621197045755809} a^{19} + \frac{5227862936360443}{11621197045755809} a^{18} + \frac{936193016224591}{11621197045755809} a^{17} + \frac{1137964186685074}{11621197045755809} a^{16} + \frac{1679709504754212}{11621197045755809} a^{15} - \frac{2317944722553866}{11621197045755809} a^{14} - \frac{3613827566560042}{11621197045755809} a^{13} + \frac{2801138704023939}{11621197045755809} a^{12} + \frac{4604014830341277}{11621197045755809} a^{11} + \frac{5537022854140037}{11621197045755809} a^{10} + \frac{3049368312068064}{11621197045755809} a^{9} - \frac{132514699271024}{11621197045755809} a^{8} + \frac{2253140937488505}{11621197045755809} a^{7} - \frac{3253293660141595}{11621197045755809} a^{6} - \frac{5313704325926755}{11621197045755809} a^{5} - \frac{311974836537642}{11621197045755809} a^{4} - \frac{3735429836988250}{11621197045755809} a^{3} - \frac{1859253110141149}{11621197045755809} a^{2} - \frac{126401018982850}{374877324056639} a + \frac{2166275284347116}{11621197045755809}$

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:  \( -\frac{4929493966726890}{11621197045755809} a^{19} + \frac{5282563030638154}{11621197045755809} a^{18} - \frac{34437343595762837}{11621197045755809} a^{17} + \frac{46243062829156822}{11621197045755809} a^{16} - \frac{143343889090408205}{11621197045755809} a^{15} + \frac{143201692265212973}{11621197045755809} a^{14} - \frac{328093617028295407}{11621197045755809} a^{13} + \frac{332490426702114339}{11621197045755809} a^{12} - \frac{768859626327396593}{11621197045755809} a^{11} + \frac{624994301352428379}{11621197045755809} a^{10} - \frac{939107803337265894}{11621197045755809} a^{9} + \frac{296991924219718267}{11621197045755809} a^{8} - \frac{473383476082622112}{11621197045755809} a^{7} + \frac{33305454427094831}{11621197045755809} a^{6} - \frac{200830672006576389}{11621197045755809} a^{5} - \frac{14718843412783821}{11621197045755809} a^{4} - \frac{70429409585194282}{11621197045755809} a^{3} + \frac{2558190389837250}{11621197045755809} a^{2} - \frac{1096559869730011}{374877324056639} a - \frac{5176828919911720}{11621197045755809} \) (order $10$)
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:  \( 92319.7736122 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T654:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 57600
The 70 conjugacy class representatives for t20n654 are not computed
Character table for t20n654 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 10.8.85992371875.1

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 $20$ $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $20$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
27517559Data not computed