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Properties

Label 20.0.12304650271...0000.4
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{34}\cdot 17^{10}$
Root discriminant $127.20$
Ramified primes $2, 5, 17$
Class number $28056248$ (GRH)
Class group $[28056248]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5962868284301, -8326865360, 2554775839555, -929158550, 515424060710, 9394408, 64418095255, 7465670, 5521535750, 549990, 339291893, 17670, 15154205, 200, 486760, -2, 10795, 0, 150, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 150*x^18 + 10795*x^16 - 2*x^15 + 486760*x^14 + 200*x^13 + 15154205*x^12 + 17670*x^11 + 339291893*x^10 + 549990*x^9 + 5521535750*x^8 + 7465670*x^7 + 64418095255*x^6 + 9394408*x^5 + 515424060710*x^4 - 929158550*x^3 + 2554775839555*x^2 - 8326865360*x + 5962868284301)
 
gp: K = bnfinit(x^20 + 150*x^18 + 10795*x^16 - 2*x^15 + 486760*x^14 + 200*x^13 + 15154205*x^12 + 17670*x^11 + 339291893*x^10 + 549990*x^9 + 5521535750*x^8 + 7465670*x^7 + 64418095255*x^6 + 9394408*x^5 + 515424060710*x^4 - 929158550*x^3 + 2554775839555*x^2 - 8326865360*x + 5962868284301, 1)
 

Normalized defining polynomial

\( x^{20} + 150 x^{18} + 10795 x^{16} - 2 x^{15} + 486760 x^{14} + 200 x^{13} + 15154205 x^{12} + 17670 x^{11} + 339291893 x^{10} + 549990 x^{9} + 5521535750 x^{8} + 7465670 x^{7} + 64418095255 x^{6} + 9394408 x^{5} + 515424060710 x^{4} - 929158550 x^{3} + 2554775839555 x^{2} - 8326865360 x + 5962868284301 \)

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:  \(1230465027129516601562500000000000000000000=2^{20}\cdot 5^{34}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $127.20$
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:  $2, 5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1700=2^{2}\cdot 5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1700}(1,·)$, $\chi_{1700}(69,·)$, $\chi_{1700}(1089,·)$, $\chi_{1700}(1291,·)$, $\chi_{1700}(271,·)$, $\chi_{1700}(1361,·)$, $\chi_{1700}(1699,·)$, $\chi_{1700}(1429,·)$, $\chi_{1700}(409,·)$, $\chi_{1700}(1359,·)$, $\chi_{1700}(1631,·)$, $\chi_{1700}(611,·)$, $\chi_{1700}(679,·)$, $\chi_{1700}(681,·)$, $\chi_{1700}(749,·)$, $\chi_{1700}(339,·)$, $\chi_{1700}(951,·)$, $\chi_{1700}(1019,·)$, $\chi_{1700}(1021,·)$, $\chi_{1700}(341,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{149} a^{18} - \frac{72}{149} a^{17} - \frac{39}{149} a^{16} + \frac{29}{149} a^{15} - \frac{31}{149} a^{14} + \frac{32}{149} a^{13} + \frac{38}{149} a^{12} + \frac{7}{149} a^{11} + \frac{59}{149} a^{10} - \frac{51}{149} a^{9} - \frac{18}{149} a^{8} - \frac{2}{149} a^{7} - \frac{5}{149} a^{6} - \frac{44}{149} a^{5} + \frac{49}{149} a^{4} - \frac{45}{149} a^{3} - \frac{38}{149} a^{2} - \frac{51}{149} a + \frac{20}{149}$, $\frac{1}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{19} + \frac{6245179826185378817036355340034379236995942970333676303649556746158614626978541}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{18} - \frac{696297370414873080061333110203535256429220078278658593107245712932232579860324952}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{17} + \frac{1081832944938775454473665462153354958758731107906632683553406257385887596780089256}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{16} - \frac{1515594028575829883150328847553081788191073202259517022516859818923048846617679238}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{15} - \frac{1111516163253297726199368369139670375950385252585359777802729749136946842350133498}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{14} + \frac{475550165839717027720173907247289475458534752658594414979687507125510111266458608}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{13} + \frac{1057034256814498544151404922330352702236044226759512674923519064706386873334179530}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{12} + \frac{480077160137439233308137075628807493514860601364254148883618841142125195473753823}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{11} - \frac{318633796297357559146762128997727712180941556496075624084860038406355043354913061}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{10} - \frac{205907182621453627317040773074379493023022816434077610556061616523503335783144122}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{9} - \frac{450121775440522467650969636349577769204578207331533326716778140258856258336212214}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{8} + \frac{1019883209691895699243367011287180445482388347367682087933390413414238031813470360}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{7} + \frac{1477346738346854747634941413181116384620907432494378997048183363494133527443470769}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{6} + \frac{242361873957709025159484401513664637137995277796892015842279433405383074450508687}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{5} - \frac{644082392496113317006432654938310745752959474053285371835235474131719343217448289}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{4} - \frac{783198968068270668267459029453891300803039180891003176376283547740789772805134764}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{3} - \frac{633005311542010881593266520843121291355694226538181712218909319827981937974034564}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a^{2} - \frac{1527093688118124317534336971191184860951858645484061694431083636565455765205116998}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057} a + \frac{351851798821843011095608412604318913356545731395332332681236876182433222772875896}{3094915019473029545912166499306290576016429838864350928204081940349036324049780057}$

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

Class group and class number

$C_{28056248}$, which has order $28056248$ (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:  \( -1 \) (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:  \( 161406.8376411007 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-85}) \), \(\Q(\sqrt{-17}) \), \(\Q(\sqrt{5}, \sqrt{-17})\), 5.5.390625.1, \(\Q(\zeta_{25})^+\), 10.0.1109263281250000000000.3, 10.0.221852656250000000000.4

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
17Data not computed