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Properties

Label 20.8.17165941082...8576.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{2}\cdot 331^{2}$
Root discriminant $289.56$
Ramified primes $2, 11, 23, 199, 331$
Class number $8$ (GRH)
Class group $[2, 2, 2]$ (GRH)
Galois group 20T331

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2350270064813056, 0, -584275946368000, 0, -623091135441152, 0, -51749801951232, 0, 99168290752, 0, 63743484160, 0, 490247200, 0, -7164256, 0, -66884, 0, 48, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 48*x^18 - 66884*x^16 - 7164256*x^14 + 490247200*x^12 + 63743484160*x^10 + 99168290752*x^8 - 51749801951232*x^6 - 623091135441152*x^4 - 584275946368000*x^2 + 2350270064813056)
 
gp: K = bnfinit(x^20 + 48*x^18 - 66884*x^16 - 7164256*x^14 + 490247200*x^12 + 63743484160*x^10 + 99168290752*x^8 - 51749801951232*x^6 - 623091135441152*x^4 - 584275946368000*x^2 + 2350270064813056, 1)
 

Normalized defining polynomial

\( x^{20} + 48 x^{18} - 66884 x^{16} - 7164256 x^{14} + 490247200 x^{12} + 63743484160 x^{10} + 99168290752 x^{8} - 51749801951232 x^{6} - 623091135441152 x^{4} - 584275946368000 x^{2} + 2350270064813056 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(17165941082918625907388507845862234378814751768576=2^{40}\cdot 11^{16}\cdot 23^{8}\cdot 199^{2}\cdot 331^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $289.56$
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:  $2, 11, 23, 199, 331$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{64} a^{11} - \frac{1}{32} a^{9} - \frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{704} a^{12} - \frac{1}{352} a^{10} + \frac{1}{88} a^{8} - \frac{1}{22} a^{6} - \frac{5}{44} a^{4} + \frac{1}{11} a^{2} + \frac{5}{11}$, $\frac{1}{1408} a^{13} - \frac{1}{704} a^{11} + \frac{1}{176} a^{9} + \frac{7}{176} a^{7} - \frac{5}{88} a^{5} - \frac{9}{44} a^{3} - \frac{3}{11} a$, $\frac{1}{1408} a^{14} + \frac{1}{352} a^{10} - \frac{1}{88} a^{8} + \frac{1}{44} a^{6} - \frac{3}{44} a^{4} - \frac{2}{11} a^{2} + \frac{5}{11}$, $\frac{1}{2816} a^{15} + \frac{1}{704} a^{11} + \frac{9}{352} a^{9} + \frac{1}{88} a^{7} + \frac{1}{11} a^{5} + \frac{7}{44} a^{3} - \frac{3}{11} a$, $\frac{1}{64768} a^{16} + \frac{1}{32384} a^{14} + \frac{1}{92} a^{10} - \frac{35}{4048} a^{8} - \frac{37}{1012} a^{6} + \frac{43}{506} a^{4} - \frac{1}{22} a^{2}$, $\frac{1}{129536} a^{17} + \frac{1}{64768} a^{15} + \frac{1}{184} a^{11} + \frac{109}{4048} a^{9} + \frac{179}{4048} a^{7} - \frac{167}{2024} a^{5} - \frac{1}{44} a^{3} - \frac{1}{2} a$, $\frac{1}{203109801718711281787176756611989099362894483975103938427012926976} a^{18} + \frac{143187479097643755900659954873709967556923636505634682641945}{50777450429677820446794189152997274840723620993775984606753231744} a^{16} + \frac{3599194358159364467473167270259855722261185556299337338652197}{12694362607419455111698547288249318710180905248443996151688307936} a^{14} + \frac{8864889326266576348636587845245276326262227057081405031043879}{25388725214838910223397094576498637420361810496887992303376615872} a^{12} + \frac{53864187957027927991229343629269955526057796789149793808472537}{12694362607419455111698547288249318710180905248443996151688307936} a^{10} + \frac{4738266056935983436365534187722180857981707099148264629183189}{1586795325927431888962318411031164838772613156055499518961038492} a^{8} - \frac{29252257934126398640896057680850173934441249874625497759707899}{1586795325927431888962318411031164838772613156055499518961038492} a^{6} - \frac{178318517598125983869139019495786451253504872053970905669646011}{1586795325927431888962318411031164838772613156055499518961038492} a^{4} + \frac{3292811841276524926897014033414334272334562417022588797402387}{17247775281819911836546939250338748247528403870168473032185201} a^{2} - \frac{41791404648635886414622194776683967384089978163247657774}{261849660414154030523416770413073649934391046929032975029}$, $\frac{1}{406219603437422563574353513223978198725788967950207876854025853952} a^{19} + \frac{143187479097643755900659954873709967556923636505634682641945}{101554900859355640893588378305994549681447241987551969213506463488} a^{17} + \frac{3599194358159364467473167270259855722261185556299337338652197}{25388725214838910223397094576498637420361810496887992303376615872} a^{15} + \frac{8864889326266576348636587845245276326262227057081405031043879}{50777450429677820446794189152997274840723620993775984606753231744} a^{13} + \frac{53864187957027927991229343629269955526057796789149793808472537}{25388725214838910223397094576498637420361810496887992303376615872} a^{11} - \frac{377745767254114038495117466006902486261226460617281821223526867}{12694362607419455111698547288249318710180905248443996151688307936} a^{9} + \frac{338194315613605174958787487396090861824270789264623884220843825}{6347181303709727555849273644124659355090452624221998075844153968} a^{7} + \frac{54595078470932997092860145815501189609912104239975993517653403}{793397662963715944481159205515582419386306578027749759480519246} a^{5} - \frac{10662151599266861982752911183510079702859279036123295437380427}{68991101127279647346187757001354992990113615480673892128740804} a^{3} - \frac{20895702324317943207311097388341983692044989081623828887}{261849660414154030523416770413073649934391046929032975029} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}$, which has order $8$ (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:  $13$
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:  \( 15271111019800000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T331:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 5120
The 56 conjugacy class representatives for t20n331 are not computed
Character table for t20n331 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2670699013250048.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$11$11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
11.10.8.5$x^{10} - 2321 x^{5} + 2033647$$5$$2$$8$$C_{10}$$[\ ]_{5}^{2}$
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.2$x^{4} - 23$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$199$199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.2.0.1$x^{2} - x + 6$$1$$2$$0$$C_2$$[\ ]^{2}$
199.4.2.2$x^{4} - 199 x^{2} + 237606$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
199.4.0.1$x^{4} - x + 3$$1$$4$$0$$C_4$$[\ ]^{4}$
331Data not computed