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Properties

Label 20.0.71560576287...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{32}\cdot 17^{15}$
Root discriminant $310.98$
Ramified primes $2, 5, 17$
Class number $213448324$ (GRH)
Class group $[2, 106724162]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11291153060857, 1107352296080, 4196964542000, 330780773040, 722807009110, 40711037368, 76757659400, 2719793140, 5668306995, 108420400, 310138170, 2552300, 12830690, 21060, 396150, -764, 8800, -20, 130, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 130*x^18 - 20*x^17 + 8800*x^16 - 764*x^15 + 396150*x^14 + 21060*x^13 + 12830690*x^12 + 2552300*x^11 + 310138170*x^10 + 108420400*x^9 + 5668306995*x^8 + 2719793140*x^7 + 76757659400*x^6 + 40711037368*x^5 + 722807009110*x^4 + 330780773040*x^3 + 4196964542000*x^2 + 1107352296080*x + 11291153060857)
 
gp: K = bnfinit(x^20 + 130*x^18 - 20*x^17 + 8800*x^16 - 764*x^15 + 396150*x^14 + 21060*x^13 + 12830690*x^12 + 2552300*x^11 + 310138170*x^10 + 108420400*x^9 + 5668306995*x^8 + 2719793140*x^7 + 76757659400*x^6 + 40711037368*x^5 + 722807009110*x^4 + 330780773040*x^3 + 4196964542000*x^2 + 1107352296080*x + 11291153060857, 1)
 

Normalized defining polynomial

\( x^{20} + 130 x^{18} - 20 x^{17} + 8800 x^{16} - 764 x^{15} + 396150 x^{14} + 21060 x^{13} + 12830690 x^{12} + 2552300 x^{11} + 310138170 x^{10} + 108420400 x^{9} + 5668306995 x^{8} + 2719793140 x^{7} + 76757659400 x^{6} + 40711037368 x^{5} + 722807009110 x^{4} + 330780773040 x^{3} + 4196964542000 x^{2} + 1107352296080 x + 11291153060857 \)

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:  \(71560576287745394825000000000000000000000000000000=2^{30}\cdot 5^{32}\cdot 17^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $310.98$
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:  \(3400=2^{3}\cdot 5^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{3400}(1,·)$, $\chi_{3400}(2571,·)$, $\chi_{3400}(2291,·)$, $\chi_{3400}(1801,·)$, $\chi_{3400}(1611,·)$, $\chi_{3400}(531,·)$, $\chi_{3400}(1361,·)$, $\chi_{3400}(1891,·)$, $\chi_{3400}(1211,·)$, $\chi_{3400}(2041,·)$, $\chi_{3400}(3161,·)$, $\chi_{3400}(2971,·)$, $\chi_{3400}(2721,·)$, $\chi_{3400}(931,·)$, $\chi_{3400}(1121,·)$, $\chi_{3400}(681,·)$, $\chi_{3400}(2481,·)$, $\chi_{3400}(3251,·)$, $\chi_{3400}(441,·)$, $\chi_{3400}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{28} a^{16} + \frac{1}{14} a^{15} + \frac{1}{7} a^{14} - \frac{3}{28} a^{13} - \frac{1}{14} a^{12} + \frac{3}{28} a^{11} - \frac{1}{4} a^{10} - \frac{1}{7} a^{9} + \frac{2}{7} a^{8} + \frac{3}{28} a^{7} - \frac{11}{28} a^{6} - \frac{9}{28} a^{5} + \frac{1}{28} a^{3} + \frac{1}{14} a^{2} + \frac{3}{28} a - \frac{1}{14}$, $\frac{1}{28} a^{17} + \frac{3}{28} a^{14} + \frac{1}{7} a^{13} - \frac{1}{4} a^{12} + \frac{1}{28} a^{11} - \frac{1}{7} a^{10} - \frac{3}{7} a^{9} + \frac{1}{28} a^{8} + \frac{11}{28} a^{7} - \frac{1}{28} a^{6} + \frac{1}{7} a^{5} + \frac{1}{28} a^{4} + \frac{13}{28} a^{2} + \frac{3}{14} a - \frac{5}{14}$, $\frac{1}{77527654291898506385061138172} a^{18} + \frac{288242617047217111941483883}{19381913572974626596265284543} a^{17} - \frac{1325574347801416558344585685}{77527654291898506385061138172} a^{16} - \frac{247951393318845825717051156}{19381913572974626596265284543} a^{15} + \frac{2693801497010088365394519744}{19381913572974626596265284543} a^{14} + \frac{2176458839192235085297974894}{19381913572974626596265284543} a^{13} + \frac{5863392815044502924560821959}{38763827145949253192530569086} a^{12} + \frac{13065683422501627392255169603}{77527654291898506385061138172} a^{11} + \frac{443808682505668082912939628}{19381913572974626596265284543} a^{10} + \frac{4706004216990826230263233609}{38763827145949253192530569086} a^{9} + \frac{14868411824625837306442380185}{77527654291898506385061138172} a^{8} + \frac{3621249223439191448591366517}{19381913572974626596265284543} a^{7} - \frac{472842153551791794908633366}{2768844796139232370895040649} a^{6} - \frac{32668242202788780413925519639}{77527654291898506385061138172} a^{5} - \frac{18917035764378526525972138989}{77527654291898506385061138172} a^{4} + \frac{331475374192335959921161681}{38763827145949253192530569086} a^{3} - \frac{589771140063154591380275613}{77527654291898506385061138172} a^{2} - \frac{1786137274049354257501059673}{11075379184556929483580162596} a - \frac{9308169569985606917376794521}{77527654291898506385061138172}$, $\frac{1}{2876472461584273384769751250368320030012006699782552542520705216028} a^{19} + \frac{2626993349560044438475990317395387261}{2876472461584273384769751250368320030012006699782552542520705216028} a^{18} - \frac{50785374953551750454920877027113550916340605429699252372257680627}{2876472461584273384769751250368320030012006699782552542520705216028} a^{17} - \frac{291230727768765796540503128674592270974854079713244726540425444}{719118115396068346192437812592080007503001674945638135630176304007} a^{16} + \frac{7192481517327667152715792421511365963480487063278796947682380697}{205462318684590956054982232169165716429429049984468038751478944002} a^{15} + \frac{314897910487717940466323523233677674772329695552847093847169460497}{1438236230792136692384875625184160015006003349891276271260352608014} a^{14} - \frac{461692578351152497352214331480959570712413367513499491493093201895}{2876472461584273384769751250368320030012006699782552542520705216028} a^{13} - \frac{541055662588922363164595774459669913434743144224760669153456768489}{2876472461584273384769751250368320030012006699782552542520705216028} a^{12} - \frac{68570363130405826595814637181747555416238824731538843051597845061}{1438236230792136692384875625184160015006003349891276271260352608014} a^{11} + \frac{244992797932232468074578403037468554362715640677664129904649640541}{2876472461584273384769751250368320030012006699782552542520705216028} a^{10} - \frac{356739747305826299425288019228534753219893994365150139674133961245}{2876472461584273384769751250368320030012006699782552542520705216028} a^{9} + \frac{918965145410328507075335415657446446393570360585177871357557228279}{2876472461584273384769751250368320030012006699782552542520705216028} a^{8} - \frac{939788958228994339605007655757201637190245633059308399269543130233}{2876472461584273384769751250368320030012006699782552542520705216028} a^{7} - \frac{433837771770122652830440481638931536072503506242307137609212237387}{1438236230792136692384875625184160015006003349891276271260352608014} a^{6} - \frac{564258725643503223557540189496325786833333627121672484358178411979}{2876472461584273384769751250368320030012006699782552542520705216028} a^{5} - \frac{752117416559595430057188448886222019353229453590147234374944933803}{2876472461584273384769751250368320030012006699782552542520705216028} a^{4} + \frac{169240790951960790287091027797848881296795580173231536176409835396}{719118115396068346192437812592080007503001674945638135630176304007} a^{3} + \frac{69141007613826368098212187448035560119899333471217282476525938125}{205462318684590956054982232169165716429429049984468038751478944002} a^{2} - \frac{125882132484940476451910219886021471458418096358714941459550122699}{2876472461584273384769751250368320030012006699782552542520705216028} a + \frac{1299762800757316206571747500778583763427278327021510849786049570411}{2876472461584273384769751250368320030012006699782552542520705216028}$

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

Class group and class number

$C_{2}\times C_{106724162}$, which has order $213448324$ (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:  \( 103447354.91828637 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

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

Intermediate fields

\(\Q(\sqrt{17}) \), 4.0.314432.2, 5.5.390625.1, 10.10.216652984619140625.1

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 $20$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ $20$ $20$ $20$ $20$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
2.10.15.13$x^{10} - 2 x^{8} - 4 x^{6} - 48 x^{2} - 96$$2$$5$$15$$C_{10}$$[3]^{5}$
5Data not computed
17Data not computed