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Properties

Label 20.10.5795258604...9008.2
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 11^{18}\cdot 1583^{5}$
Root discriminant $109.18$
Ramified primes $2, 11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9281006867, -5789595834, -1107249165, -321339590, 2300571482, 1258865740, 350417694, -37548434, -170875749, -40521162, -6849787, -2159284, 1931801, 310776, -55563, 4368, -2008, -250, 26, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 26*x^18 - 250*x^17 - 2008*x^16 + 4368*x^15 - 55563*x^14 + 310776*x^13 + 1931801*x^12 - 2159284*x^11 - 6849787*x^10 - 40521162*x^9 - 170875749*x^8 - 37548434*x^7 + 350417694*x^6 + 1258865740*x^5 + 2300571482*x^4 - 321339590*x^3 - 1107249165*x^2 - 5789595834*x - 9281006867)
 
gp: K = bnfinit(x^20 - 2*x^19 + 26*x^18 - 250*x^17 - 2008*x^16 + 4368*x^15 - 55563*x^14 + 310776*x^13 + 1931801*x^12 - 2159284*x^11 - 6849787*x^10 - 40521162*x^9 - 170875749*x^8 - 37548434*x^7 + 350417694*x^6 + 1258865740*x^5 + 2300571482*x^4 - 321339590*x^3 - 1107249165*x^2 - 5789595834*x - 9281006867, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 26 x^{18} - 250 x^{17} - 2008 x^{16} + 4368 x^{15} - 55563 x^{14} + 310776 x^{13} + 1931801 x^{12} - 2159284 x^{11} - 6849787 x^{10} - 40521162 x^{9} - 170875749 x^{8} - 37548434 x^{7} + 350417694 x^{6} + 1258865740 x^{5} + 2300571482 x^{4} - 321339590 x^{3} - 1107249165 x^{2} - 5789595834 x - 9281006867 \)

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:  \(-57952586042902321051693895129329600299008=-\,2^{20}\cdot 11^{18}\cdot 1583^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.18$
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:  $2, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{14} - \frac{4}{11} a^{13} + \frac{3}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{10}$, $\frac{1}{11} a^{16} - \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{5}{11} a^{12} + \frac{2}{11} a^{11} - \frac{1}{11} a^{10}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{14} - \frac{3}{11} a^{13} - \frac{5}{11} a^{12} + \frac{3}{11} a^{11} - \frac{5}{11} a^{10}$, $\frac{1}{11} a^{18} + \frac{2}{11} a^{14} + \frac{4}{11} a^{13} - \frac{1}{11} a^{12} + \frac{2}{11} a^{11} + \frac{5}{11} a^{10}$, $\frac{1}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{19} - \frac{9531776812925534979179357347111040861680846911301042472999308673184169490688263007403120386215121125}{237962620588058560361546131214265635984501123431743167652895282784031910776392474269825489766617671667} a^{18} + \frac{23795553345297025435716864312481407212967768094946837838681401067147706734135557092281540693323491186}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{17} + \frac{12166296227891080811982395510961872499680881726405915716938141642942112918772901014630270885034800439}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{16} + \frac{184484225233243662184005593626335101988244702195255249102888143657427780253975463009157751127373545173}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{15} + \frac{2212426073538246229094164736721131352807362561689114499285795505060340057792659530996483466169908868732}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{14} + \frac{2596273141328011662643915234022999396690644121529103416021677315583855945701740061028185816899798861442}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{13} + \frac{883919052169846600457516249281165626431580548284946740632752743235458993299864225639599140921674124360}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{12} + \frac{218062050249743772394410333743113845979019308734368447286258079163316392875535513528441321683114108127}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{11} + \frac{847519833165037684542209820504593182122168246972197029890211661410189558335470424628189429025037923143}{5473140273525346888315561017928109627643525838930092856016591504032733947857026908205986264632206448341} a^{10} - \frac{104119905172908830850850532574963094365902039407603320961493083820468470457469493991537768524513732727}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{9} + \frac{239646053103430109783707743019832004994590243757230222885579583337448675120266856151185987462854468283}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{8} + \frac{26599162264274817774893104567686525253086044770891928070041708512496408502142773170175617843371233394}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{7} - \frac{119026923575522944789834354584093055166855751192834125596278033021678665333256665417860601936797477033}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{6} + \frac{29245320798290363354135314020789258749977169870180901272982271202964389562473500411783233919465292455}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{5} - \frac{237132125597836567578169650606539598227388800524919482087540400094064359898890166618739395111439687175}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{4} + \frac{156126089692013799736233456063019871085786051615196955388241305546385401494826459429146038495788881531}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{3} + \frac{116111263738391412310026262234667255355980173210068548426732544354225479123187458798071934543973285343}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a^{2} - \frac{236551774177644075885407373729022135237869731631597045815128441729831028911525368523744392072488098061}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031} a - \frac{211420388579667291281821526105879541266548022808363201886858283401308115060985202644006837533352999773}{497558206684122444392323728902555420694865985357281168728781045821157631623366082564180569512018768031}$

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:  \( 8983236423350 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
2Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed