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Properties

Label 20.0.21333423461...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{35}\cdot 31^{18}$
Root discriminant $735.28$
Ramified primes $2, 5, 31$
Class number $6208336000$ (GRH)
Class group $[2, 2, 2, 2, 2, 10, 19401050]$ (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![26243954842880, 0, 2012400282092000, 0, 662889174664625, 0, 71713231838300, 0, 3798910889050, 0, 114603766080, 0, 2101010275, 0, 23832800, 0, 163370, 0, 620, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 620*x^18 + 163370*x^16 + 23832800*x^14 + 2101010275*x^12 + 114603766080*x^10 + 3798910889050*x^8 + 71713231838300*x^6 + 662889174664625*x^4 + 2012400282092000*x^2 + 26243954842880)
 
gp: K = bnfinit(x^20 + 620*x^18 + 163370*x^16 + 23832800*x^14 + 2101010275*x^12 + 114603766080*x^10 + 3798910889050*x^8 + 71713231838300*x^6 + 662889174664625*x^4 + 2012400282092000*x^2 + 26243954842880, 1)
 

Normalized defining polynomial

\( x^{20} + 620 x^{18} + 163370 x^{16} + 23832800 x^{14} + 2101010275 x^{12} + 114603766080 x^{10} + 3798910889050 x^{8} + 71713231838300 x^{6} + 662889174664625 x^{4} + 2012400282092000 x^{2} + 26243954842880 \)

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:  \(2133342346188491938901276370239257812500000000000000000000=2^{20}\cdot 5^{35}\cdot 31^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $735.28$
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:  $2, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3100=2^{2}\cdot 5^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{3100}(1,·)$, $\chi_{3100}(387,·)$, $\chi_{3100}(2389,·)$, $\chi_{3100}(263,·)$, $\chi_{3100}(2761,·)$, $\chi_{3100}(2323,·)$, $\chi_{3100}(2581,·)$, $\chi_{3100}(2329,·)$, $\chi_{3100}(1883,·)$, $\chi_{3100}(221,·)$, $\chi_{3100}(1827,·)$, $\chi_{3100}(2341,·)$, $\chi_{3100}(743,·)$, $\chi_{3100}(647,·)$, $\chi_{3100}(109,·)$, $\chi_{3100}(2107,·)$, $\chi_{3100}(969,·)$, $\chi_{3100}(249,·)$, $\chi_{3100}(3003,·)$, $\chi_{3100}(767,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{3}{8} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{3}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{2} a^{4} - \frac{3}{8} a^{2}$, $\frac{1}{8} a^{9} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{38709824} a^{10} + \frac{5}{624352} a^{8} + \frac{1085}{1248704} a^{6} + \frac{24025}{624352} a^{4} - \frac{503929}{1248704} a^{2} - \frac{29201}{78044}$, $\frac{1}{38709824} a^{11} + \frac{5}{624352} a^{9} + \frac{1085}{1248704} a^{7} + \frac{24025}{624352} a^{5} - \frac{503929}{1248704} a^{3} - \frac{29201}{78044} a$, $\frac{1}{38709824} a^{12} - \frac{2015}{1248704} a^{8} + \frac{5969}{312176} a^{6} + \frac{521547}{1248704} a^{4} - \frac{12525}{624352} a^{2} - \frac{397}{39022}$, $\frac{1}{38709824} a^{13} - \frac{2015}{1248704} a^{9} + \frac{5969}{312176} a^{7} - \frac{102805}{1248704} a^{5} + \frac{299651}{624352} a^{3} + \frac{9557}{19511} a$, $\frac{1}{1200004544} a^{14} + \frac{5425}{624352} a^{8} + \frac{401}{39022} a^{6} + \frac{234164}{604841} a^{4} - \frac{387151}{1248704} a^{2} + \frac{5161}{78044}$, $\frac{1}{715202708224} a^{15} + \frac{15}{23071055104} a^{13} - \frac{95}{11535527552} a^{11} - \frac{21275}{744227584} a^{9} - \frac{1400425}{372113792} a^{7} - \frac{978738065}{23071055104} a^{5} + \frac{68916913}{744227584} a^{3} + \frac{17716305}{46514224} a$, $\frac{1}{20913957593886208} a^{16} - \frac{964459}{20913957593886208} a^{14} + \frac{1961043}{337321896675584} a^{12} - \frac{2482689}{674643793351168} a^{10} - \frac{9196922015}{10881351505664} a^{8} - \frac{39754422986613}{674643793351168} a^{6} + \frac{291085258705019}{674643793351168} a^{4} + \frac{1314556079837}{2720337876416} a^{2} - \frac{401827453}{1141081324}$, $\frac{1}{648332685410472448} a^{17} + \frac{17}{20913957593886208} a^{15} + \frac{1834965}{168660948337792} a^{13} - \frac{4870115}{674643793351168} a^{11} - \frac{3709596825}{5440675752832} a^{9} - \frac{473555757204145}{20913957593886208} a^{7} - \frac{30021187820097}{674643793351168} a^{5} - \frac{2460152957209}{10881351505664} a^{3} + \frac{265374756075}{680084469104} a$, $\frac{1}{648332685410472448} a^{18} - \frac{63785}{20913957593886208} a^{14} - \frac{1832649}{674643793351168} a^{12} + \frac{670125}{674643793351168} a^{10} - \frac{1046290434061411}{20913957593886208} a^{8} - \frac{9102666666085}{168660948337792} a^{6} - \frac{278273413612745}{674643793351168} a^{4} + \frac{55877875605}{1360168938208} a^{2} - \frac{130038196}{285270331}$, $\frac{1}{648332685410472448} a^{19} - \frac{171}{674643793351168} a^{15} - \frac{30819}{21762703011328} a^{13} + \frac{6986397}{674643793351168} a^{11} - \frac{7028984363859}{140362131502592} a^{9} - \frac{10225605083605}{168660948337792} a^{7} + \frac{895748033709}{21762703011328} a^{5} + \frac{1755835682927}{5440675752832} a^{3} + \frac{820196829}{1899677288} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{19401050}$, which has order $6208336000$ (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:  \( 127725981510.55186 \) (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

\(\Q(\sqrt{5}) \), 4.0.1922000.2, 5.5.360750390625.4, 10.10.650704221680450439453125.4

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 $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ R $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.1.0.1}{1} }^{20}$

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
$2$2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
2.4.4.2$x^{4} - x^{2} + 5$$2$$2$$4$$C_4$$[2]^{2}$
5Data not computed
$31$31.10.9.4$x^{10} - 3647119$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.9.4$x^{10} - 3647119$$10$$1$$9$$C_{10}$$[\ ]_{10}$