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Properties

Label 20.0.44515812477...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{10}\cdot 181^{18}$
Root discriminant $481.31$
Ramified primes $2, 5, 181$
Class number $1418455320$ (GRH)
Class group $[2, 709227660]$ (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![6218998361089, -1475502846552, -111289384821, -135889082190, 51009829022, 42036545916, -8933825593, -5617531168, 1312082182, 422781594, -109533967, -21730684, 6380662, 753552, -248884, -17866, 6709, 260, -111, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 111*x^18 + 260*x^17 + 6709*x^16 - 17866*x^15 - 248884*x^14 + 753552*x^13 + 6380662*x^12 - 21730684*x^11 - 109533967*x^10 + 422781594*x^9 + 1312082182*x^8 - 5617531168*x^7 - 8933825593*x^6 + 42036545916*x^5 + 51009829022*x^4 - 135889082190*x^3 - 111289384821*x^2 - 1475502846552*x + 6218998361089)
 
gp: K = bnfinit(x^20 - 2*x^19 - 111*x^18 + 260*x^17 + 6709*x^16 - 17866*x^15 - 248884*x^14 + 753552*x^13 + 6380662*x^12 - 21730684*x^11 - 109533967*x^10 + 422781594*x^9 + 1312082182*x^8 - 5617531168*x^7 - 8933825593*x^6 + 42036545916*x^5 + 51009829022*x^4 - 135889082190*x^3 - 111289384821*x^2 - 1475502846552*x + 6218998361089, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 111 x^{18} + 260 x^{17} + 6709 x^{16} - 17866 x^{15} - 248884 x^{14} + 753552 x^{13} + 6380662 x^{12} - 21730684 x^{11} - 109533967 x^{10} + 422781594 x^{9} + 1312082182 x^{8} - 5617531168 x^{7} - 8933825593 x^{6} + 42036545916 x^{5} + 51009829022 x^{4} - 135889082190 x^{3} - 111289384821 x^{2} - 1475502846552 x + 6218998361089 \)

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:  \(445158124777785892479889640032765008613795840000000000=2^{20}\cdot 5^{10}\cdot 181^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $481.31$
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:  $2, 5, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3620=2^{2}\cdot 5\cdot 181\)
Dirichlet character group:    $\lbrace$$\chi_{3620}(1,·)$, $\chi_{3620}(1221,·)$, $\chi_{3620}(961,·)$, $\chi_{3620}(2761,·)$, $\chi_{3620}(139,·)$, $\chi_{3620}(3021,·)$, $\chi_{3620}(2659,·)$, $\chi_{3620}(599,·)$, $\chi_{3620}(3481,·)$, $\chi_{3620}(859,·)$, $\chi_{3620}(2399,·)$, $\chi_{3620}(3619,·)$, $\chi_{3620}(421,·)$, $\chi_{3620}(3561,·)$, $\chi_{3620}(3119,·)$, $\chi_{3620}(59,·)$, $\chi_{3620}(501,·)$, $\chi_{3620}(361,·)$, $\chi_{3620}(3259,·)$, $\chi_{3620}(3199,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{43} a^{16} + \frac{10}{43} a^{15} + \frac{16}{43} a^{14} + \frac{12}{43} a^{13} + \frac{6}{43} a^{12} + \frac{2}{43} a^{11} + \frac{21}{43} a^{10} - \frac{13}{43} a^{9} + \frac{7}{43} a^{8} - \frac{19}{43} a^{7} - \frac{18}{43} a^{6} - \frac{1}{43} a^{5} - \frac{16}{43} a^{4} - \frac{19}{43} a^{3} + \frac{11}{43} a^{2} + \frac{2}{43} a - \frac{2}{43}$, $\frac{1}{731} a^{17} - \frac{6}{731} a^{16} - \frac{144}{731} a^{15} - \frac{115}{731} a^{14} + \frac{287}{731} a^{13} + \frac{78}{731} a^{12} + \frac{32}{731} a^{11} - \frac{91}{731} a^{10} + \frac{3}{17} a^{9} - \frac{45}{731} a^{8} - \frac{359}{731} a^{7} + \frac{158}{731} a^{6} + \frac{280}{731} a^{4} - \frac{29}{731} a^{3} + \frac{256}{731} a^{2} - \frac{335}{731} a - \frac{183}{731}$, $\frac{1}{4038882258037151} a^{18} + \frac{12602171410}{55327154219687} a^{17} + \frac{1707540706657}{576983179719593} a^{16} + \frac{529451230427480}{4038882258037151} a^{15} + \frac{1536716840112046}{4038882258037151} a^{14} - \frac{100277393059017}{237581309296303} a^{13} + \frac{1819070247668614}{4038882258037151} a^{12} + \frac{384703218143336}{4038882258037151} a^{11} - \frac{1959103513414258}{4038882258037151} a^{10} - \frac{835834795883369}{4038882258037151} a^{9} - \frac{898170143809046}{4038882258037151} a^{8} - \frac{1915271082767837}{4038882258037151} a^{7} + \frac{1749332899591538}{4038882258037151} a^{6} + \frac{240871305291258}{4038882258037151} a^{5} - \frac{412746724835870}{4038882258037151} a^{4} + \frac{1919843600968495}{4038882258037151} a^{3} - \frac{1656507891989716}{4038882258037151} a^{2} + \frac{1450695772564706}{4038882258037151} a - \frac{1542954835866894}{4038882258037151}$, $\frac{1}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{19} - \frac{2886465650515106343478861755654071283195080746676359684952073927533}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{18} - \frac{14200337010153646427089506709603006487837285607961056803008256635543283016056714}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{17} + \frac{11246890459682348757627859188773295217469419905059794633955163480846470359426321}{1637884368191411202873449412975823369945661415299049269054263610757074356261569733} a^{16} + \frac{4264189702851375950183597131632798596405781875536978234324382878160195663437492332}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{15} - \frac{3852724467682616094072743285309675816052642608169176698558169078481281508054794533}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{14} - \frac{2023798029651806810617167967886344465413097344938557405039795924254500566565759336}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{13} + \frac{648784927134902893425322367736334273282294084012766782517183758986588957917095807}{4445686142233830407799362692362949146995366698668848016004429800626344681281403561} a^{12} + \frac{2303206742598425753797433589666225120839152276541601809142862326051465942896758037}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{11} - \frac{1667436003523761062392235985276588836370898929567930782128039512943086030768277456}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{10} + \frac{1306431466231648694035620455961105431060384022019204474177479100922477679982737695}{4445686142233830407799362692362949146995366698668848016004429800626344681281403561} a^{9} - \frac{15058724174188900968291453765297989880566362811721531590953791057061889235540984847}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{8} - \frac{9877300537268355156937147241913398287888653397328652059477076190183838395301469696}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{7} - \frac{7057228110570387459308706125276209359137349897243704694716541951117955922539483249}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{6} - \frac{11124743569703862288807699334685795387249305758506113066263172723460488238769260120}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{5} - \frac{15524840271072935235057635831099319236626303587649679425863461095355787039909269479}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{4} + \frac{3811022500317563037254650657232336448800037675877183434136080666870684360123361939}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{3} + \frac{1866366935801106838082253535552814157987839847850969555156171446027523000359197464}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a^{2} + \frac{7033715373798914879856575361953321936785054619310656724346881167702252957140292668}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927} a + \frac{2844586672343965275261627141712932570428972814442082567974799801694586993949921176}{31119802995636812854595538846540644028967566890681936112031008604384412768969824927}$

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

Class group and class number

$C_{2}\times C_{709227660}$, which has order $1418455320$ (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:  \( 2738636751.4469314 \) (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{181}) \), \(\Q(\sqrt{-905}) \), \(\Q(\sqrt{-5}, \sqrt{181})\), 5.5.1073283121.1, 10.0.3686197305035202051200000.1, 10.10.208500535066053616021.1, 10.0.667201712211371571267200000.1

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 ${\href{/LocalNumberField/3.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$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}$
$181$181.10.9.1$x^{10} - 181$$10$$1$$9$$C_{10}$$[\ ]_{10}$
181.10.9.1$x^{10} - 181$$10$$1$$9$$C_{10}$$[\ ]_{10}$