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Properties

Label 20.0.21114116905...6201.2
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{10}\cdot 13^{10}$
Root discriminant $20.71$
Ramified primes $3, 11, 13$
Class number $4$
Class group $[2, 2]$
Galois group $D_{10}$ (as 20T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -375, -225, 180, 324, -1533, 2270, -1311, 53, -174, 1354, -2190, 1939, -1062, 338, -30, -16, -6, 13, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 16*x^16 - 30*x^15 + 338*x^14 - 1062*x^13 + 1939*x^12 - 2190*x^11 + 1354*x^10 - 174*x^9 + 53*x^8 - 1311*x^7 + 2270*x^6 - 1533*x^5 + 324*x^4 + 180*x^3 - 225*x^2 - 375*x + 625)
 
gp: K = bnfinit(x^20 - 6*x^19 + 13*x^18 - 6*x^17 - 16*x^16 - 30*x^15 + 338*x^14 - 1062*x^13 + 1939*x^12 - 2190*x^11 + 1354*x^10 - 174*x^9 + 53*x^8 - 1311*x^7 + 2270*x^6 - 1533*x^5 + 324*x^4 + 180*x^3 - 225*x^2 - 375*x + 625, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 13 x^{18} - 6 x^{17} - 16 x^{16} - 30 x^{15} + 338 x^{14} - 1062 x^{13} + 1939 x^{12} - 2190 x^{11} + 1354 x^{10} - 174 x^{9} + 53 x^{8} - 1311 x^{7} + 2270 x^{6} - 1533 x^{5} + 324 x^{4} + 180 x^{3} - 225 x^{2} - 375 x + 625 \)

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:  \(211141169056178733575976201=3^{10}\cdot 11^{10}\cdot 13^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.71$
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:  $3, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{5} a^{14} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{65} a^{15} + \frac{1}{65} a^{14} + \frac{28}{65} a^{13} + \frac{24}{65} a^{12} - \frac{14}{65} a^{11} + \frac{32}{65} a^{10} + \frac{1}{65} a^{9} + \frac{2}{5} a^{8} + \frac{16}{65} a^{7} - \frac{11}{65} a^{6} - \frac{1}{5} a^{5} + \frac{21}{65} a^{4} - \frac{29}{65} a^{3} + \frac{2}{5} a^{2} - \frac{6}{65} a - \frac{5}{13}$, $\frac{1}{65} a^{16} + \frac{1}{65} a^{14} - \frac{4}{65} a^{13} + \frac{14}{65} a^{12} + \frac{4}{13} a^{11} - \frac{31}{65} a^{10} - \frac{27}{65} a^{9} + \frac{16}{65} a^{8} - \frac{14}{65} a^{7} + \frac{24}{65} a^{6} - \frac{31}{65} a^{5} + \frac{28}{65} a^{4} + \frac{16}{65} a^{3} - \frac{19}{65} a^{2} + \frac{7}{65} a + \frac{5}{13}$, $\frac{1}{65} a^{17} - \frac{1}{13} a^{14} - \frac{14}{65} a^{13} - \frac{4}{65} a^{12} - \frac{17}{65} a^{11} + \frac{6}{65} a^{10} + \frac{3}{13} a^{9} + \frac{5}{13} a^{8} + \frac{8}{65} a^{7} - \frac{4}{13} a^{6} - \frac{24}{65} a^{5} - \frac{1}{13} a^{4} + \frac{2}{13} a^{3} - \frac{19}{65} a^{2} + \frac{31}{65} a + \frac{5}{13}$, $\frac{1}{392275} a^{18} + \frac{839}{392275} a^{17} - \frac{1782}{392275} a^{16} + \frac{158}{30175} a^{15} + \frac{6689}{392275} a^{14} - \frac{6217}{15691} a^{13} + \frac{93713}{392275} a^{12} + \frac{155498}{392275} a^{11} - \frac{74801}{392275} a^{10} + \frac{29428}{78455} a^{9} - \frac{5038}{23075} a^{8} + \frac{11162}{30175} a^{7} + \frac{96048}{392275} a^{6} - \frac{145326}{392275} a^{5} - \frac{441}{1207} a^{4} + \frac{143267}{392275} a^{3} + \frac{97989}{392275} a^{2} - \frac{30873}{78455} a + \frac{6293}{15691}$, $\frac{1}{2030822391196625} a^{19} + \frac{2204202719}{2030822391196625} a^{18} - \frac{10305709977237}{2030822391196625} a^{17} + \frac{194639490569}{2030822391196625} a^{16} - \frac{9029410863566}{2030822391196625} a^{15} + \frac{29495367113709}{406164478239325} a^{14} + \frac{709663116835238}{2030822391196625} a^{13} + \frac{22602314802989}{119460140658625} a^{12} - \frac{9693383880834}{70028358317125} a^{11} - \frac{43860881461573}{406164478239325} a^{10} + \frac{934732447113429}{2030822391196625} a^{9} + \frac{974587856738326}{2030822391196625} a^{8} - \frac{341078763423147}{2030822391196625} a^{7} - \frac{731361133313561}{2030822391196625} a^{6} + \frac{2868812121636}{14005671663425} a^{5} - \frac{375728181680458}{2030822391196625} a^{4} - \frac{3181841825181}{9189241589125} a^{3} - \frac{139607868435259}{406164478239325} a^{2} + \frac{26478752695666}{81232895647865} a + \frac{5558459002310}{16246579129573}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:  \( \frac{751207494}{129425937875} a^{19} - \frac{3076385969}{129425937875} a^{18} + \frac{3729437702}{129425937875} a^{17} + \frac{2291332221}{129425937875} a^{16} - \frac{4572537549}{129425937875} a^{15} - \frac{7243808758}{25885187575} a^{14} + \frac{183182098722}{129425937875} a^{13} - \frac{437142038643}{129425937875} a^{12} + \frac{21765110694}{4462963375} a^{11} - \frac{114695511156}{25885187575} a^{10} + \frac{325572347651}{129425937875} a^{9} - \frac{187972857276}{129425937875} a^{8} + \frac{32659634029}{9955841375} a^{7} - \frac{734617868474}{129425937875} a^{6} + \frac{310158271}{68660975} a^{5} - \frac{247912552452}{129425937875} a^{4} + \frac{75303325421}{129425937875} a^{3} + \frac{213915373}{1035407503} a^{2} + \frac{1528436658}{1035407503} a - \frac{1182908286}{1035407503} \) (order $6$)
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 90444.1905101 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_{10}$ (as 20T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $D_{10}$
Character table for $D_{10}$

Intermediate fields

\(\Q(\sqrt{-143}) \), \(\Q(\sqrt{429}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-143})\), 5.1.20449.1 x5, 10.0.59797108943.2, 10.2.14530697473149.1 x5, 10.0.101613269043.1 x5

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 10 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ R R ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$13$13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$