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Properties

Label 20.0.28450861599...000.13
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{16}$
Root discriminant $74.59$
Ramified primes $2, 3, 5, 11$
Class number $165500$ (GRH)
Class group $[5, 5, 6620]$ (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![1407282031, -48899262, 1158440409, -55070354, 449436464, -28741094, 108267719, -8559240, 18065990, -1614556, 2213089, -205852, 205958, -18672, 14818, -1282, 827, -66, 35, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 35*x^18 - 66*x^17 + 827*x^16 - 1282*x^15 + 14818*x^14 - 18672*x^13 + 205958*x^12 - 205852*x^11 + 2213089*x^10 - 1614556*x^9 + 18065990*x^8 - 8559240*x^7 + 108267719*x^6 - 28741094*x^5 + 449436464*x^4 - 55070354*x^3 + 1158440409*x^2 - 48899262*x + 1407282031)
 
gp: K = bnfinit(x^20 - 2*x^19 + 35*x^18 - 66*x^17 + 827*x^16 - 1282*x^15 + 14818*x^14 - 18672*x^13 + 205958*x^12 - 205852*x^11 + 2213089*x^10 - 1614556*x^9 + 18065990*x^8 - 8559240*x^7 + 108267719*x^6 - 28741094*x^5 + 449436464*x^4 - 55070354*x^3 + 1158440409*x^2 - 48899262*x + 1407282031, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 35 x^{18} - 66 x^{17} + 827 x^{16} - 1282 x^{15} + 14818 x^{14} - 18672 x^{13} + 205958 x^{12} - 205852 x^{11} + 2213089 x^{10} - 1614556 x^{9} + 18065990 x^{8} - 8559240 x^{7} + 108267719 x^{6} - 28741094 x^{5} + 449436464 x^{4} - 55070354 x^{3} + 1158440409 x^{2} - 48899262 x + 1407282031 \)

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:  \(28450861599572073223437680640000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $74.59$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1320=2^{3}\cdot 3\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{1320}(1,·)$, $\chi_{1320}(389,·)$, $\chi_{1320}(961,·)$, $\chi_{1320}(841,·)$, $\chi_{1320}(269,·)$, $\chi_{1320}(529,·)$, $\chi_{1320}(1301,·)$, $\chi_{1320}(889,·)$, $\chi_{1320}(221,·)$, $\chi_{1320}(581,·)$, $\chi_{1320}(289,·)$, $\chi_{1320}(1061,·)$, $\chi_{1320}(169,·)$, $\chi_{1320}(749,·)$, $\chi_{1320}(1181,·)$, $\chi_{1320}(49,·)$, $\chi_{1320}(361,·)$, $\chi_{1320}(1081,·)$, $\chi_{1320}(509,·)$, $\chi_{1320}(1109,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} + \frac{4}{11} a^{14} + \frac{5}{11} a^{13} + \frac{2}{11} a^{12} - \frac{4}{11} a^{11} + \frac{3}{11} a^{9} + \frac{5}{11} a^{8} + \frac{1}{11} a^{7} - \frac{2}{11} a^{6} - \frac{5}{11} a^{4} - \frac{1}{11} a^{3} + \frac{1}{11} a^{2} + \frac{1}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{16} + \frac{4}{11} a^{13} - \frac{1}{11} a^{12} + \frac{5}{11} a^{11} + \frac{3}{11} a^{10} + \frac{4}{11} a^{9} + \frac{3}{11} a^{8} + \frac{5}{11} a^{7} - \frac{3}{11} a^{6} - \frac{5}{11} a^{5} - \frac{3}{11} a^{4} + \frac{5}{11} a^{3} - \frac{3}{11} a^{2} - \frac{3}{11} a - \frac{4}{11}$, $\frac{1}{11} a^{17} + \frac{4}{11} a^{14} - \frac{1}{11} a^{13} + \frac{5}{11} a^{12} + \frac{3}{11} a^{11} + \frac{4}{11} a^{10} + \frac{3}{11} a^{9} + \frac{5}{11} a^{8} - \frac{3}{11} a^{7} - \frac{5}{11} a^{6} - \frac{3}{11} a^{5} + \frac{5}{11} a^{4} - \frac{3}{11} a^{3} - \frac{3}{11} a^{2} - \frac{4}{11} a$, $\frac{1}{3641} a^{18} - \frac{131}{3641} a^{17} + \frac{107}{3641} a^{16} - \frac{90}{3641} a^{15} + \frac{672}{3641} a^{14} + \frac{512}{3641} a^{13} + \frac{615}{3641} a^{12} + \frac{1468}{3641} a^{11} - \frac{1509}{3641} a^{10} - \frac{160}{331} a^{9} + \frac{1261}{3641} a^{8} + \frac{202}{3641} a^{7} + \frac{1333}{3641} a^{6} + \frac{237}{3641} a^{5} + \frac{1625}{3641} a^{4} - \frac{246}{3641} a^{3} - \frac{1632}{3641} a^{2} - \frac{1585}{3641} a + \frac{1469}{3641}$, $\frac{1}{23111652641309955118691787305000428681229311619322198378280275719} a^{19} - \frac{2068566632644877760767177618478446392696949737576934840060805}{23111652641309955118691787305000428681229311619322198378280275719} a^{18} + \frac{742550022281898294338365702374553388724337885707912109942374964}{23111652641309955118691787305000428681229311619322198378280275719} a^{17} + \frac{70159074657176428646943092719736299848447807384438682067999682}{23111652641309955118691787305000428681229311619322198378280275719} a^{16} - \frac{126652852350940206008291937033058022451656590582512527963215068}{23111652641309955118691787305000428681229311619322198378280275719} a^{15} + \frac{10653271571362802204059567731285570884182746303060300140227988630}{23111652641309955118691787305000428681229311619322198378280275719} a^{14} + \frac{1907455662814748229806313482681036529012880708940955062232216333}{23111652641309955118691787305000428681229311619322198378280275719} a^{13} + \frac{7304643381841342457474743415209105785092524713406156881166534394}{23111652641309955118691787305000428681229311619322198378280275719} a^{12} - \frac{10440020099889622960853740914647607137500158998109249649780259746}{23111652641309955118691787305000428681229311619322198378280275719} a^{11} - \frac{6991588362605215001144044493814462826465854753931723004633317582}{23111652641309955118691787305000428681229311619322198378280275719} a^{10} - \frac{6853959154116335057764957552287643536536185810229555934359224209}{23111652641309955118691787305000428681229311619322198378280275719} a^{9} + \frac{1994955393051283410940391102058580771164371617511782584958113157}{23111652641309955118691787305000428681229311619322198378280275719} a^{8} + \frac{3149550552415892176613477010729559839484251266252844764799216391}{23111652641309955118691787305000428681229311619322198378280275719} a^{7} - \frac{5037295014233100479141500881407023365762803281746303122042321201}{23111652641309955118691787305000428681229311619322198378280275719} a^{6} - \frac{6853202891603009746214624465043382422943477768145509575058666907}{23111652641309955118691787305000428681229311619322198378280275719} a^{5} + \frac{688966937887258512785340054625587295797740429025436951000285403}{23111652641309955118691787305000428681229311619322198378280275719} a^{4} - \frac{456657816418954878967050673993182852270380368414030583904245687}{23111652641309955118691787305000428681229311619322198378280275719} a^{3} - \frac{33214108962648112188013663908213277446623303967270005341973677}{116138957996532437782370790477389088850398550850865318483820481} a^{2} + \frac{8209105831939350527575879421253698913954577584078200013201848615}{23111652641309955118691787305000428681229311619322198378280275719} a - \frac{51257318879847212719700133551562760174760946162028257756045465}{116138957996532437782370790477389088850398550850865318483820481}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{6620}$, which has order $165500$ (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:  \( 140644.599182 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{5}, \sqrt{-6})\), \(\Q(\zeta_{11})^+\), 10.10.669871503125.1, 10.0.1706859170463744.1, 10.0.5333934907699200000.1

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
3Data not computed
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$