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Properties

Label 20.0.66362369058...000.22
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $309.81$
Ramified primes $2, 3, 5, 41$
Class number $383397888$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 2, 2, 4, 374412]$ (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![10934919736011801, -58397079968928, 1868945223446574, -39603127634430, 146248193880223, -4669719511778, 7133505512379, -255366343868, 245568378220, -8349870828, 6272846668, -184638728, 121922757, -2862362, 1839946, -32382, 21349, -314, 183, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 183*x^18 - 314*x^17 + 21349*x^16 - 32382*x^15 + 1839946*x^14 - 2862362*x^13 + 121922757*x^12 - 184638728*x^11 + 6272846668*x^10 - 8349870828*x^9 + 245568378220*x^8 - 255366343868*x^7 + 7133505512379*x^6 - 4669719511778*x^5 + 146248193880223*x^4 - 39603127634430*x^3 + 1868945223446574*x^2 - 58397079968928*x + 10934919736011801)
 
gp: K = bnfinit(x^20 - 2*x^19 + 183*x^18 - 314*x^17 + 21349*x^16 - 32382*x^15 + 1839946*x^14 - 2862362*x^13 + 121922757*x^12 - 184638728*x^11 + 6272846668*x^10 - 8349870828*x^9 + 245568378220*x^8 - 255366343868*x^7 + 7133505512379*x^6 - 4669719511778*x^5 + 146248193880223*x^4 - 39603127634430*x^3 + 1868945223446574*x^2 - 58397079968928*x + 10934919736011801, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 183 x^{18} - 314 x^{17} + 21349 x^{16} - 32382 x^{15} + 1839946 x^{14} - 2862362 x^{13} + 121922757 x^{12} - 184638728 x^{11} + 6272846668 x^{10} - 8349870828 x^{9} + 245568378220 x^{8} - 255366343868 x^{7} + 7133505512379 x^{6} - 4669719511778 x^{5} + 146248193880223 x^{4} - 39603127634430 x^{3} + 1868945223446574 x^{2} - 58397079968928 x + 10934919736011801 \)

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:  \(66362369058556182102889507973449950167040000000000=2^{30}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $309.81$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(1349,·)$, $\chi_{4920}(701,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(2189,·)$, $\chi_{4920}(4369,·)$, $\chi_{4920}(3221,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(409,·)$, $\chi_{4920}(2789,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(3749,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(4781,·)$, $\chi_{4920}(4541,·)$, $\chi_{4920}(1009,·)$, $\chi_{4920}(1849,·)$, $\chi_{4920}(769,·)$, $\chi_{4920}(2429,·)$, $\chi_{4920}(4181,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{45} a^{10} - \frac{1}{45} a^{9} + \frac{2}{15} a^{8} - \frac{2}{15} a^{7} + \frac{1}{15} a^{6} - \frac{4}{15} a^{5} + \frac{4}{9} a^{4} - \frac{11}{45} a^{3} + \frac{2}{15} a^{2} - \frac{2}{15} a + \frac{1}{5}$, $\frac{1}{45} a^{11} - \frac{1}{15} a^{7} + \frac{2}{15} a^{6} - \frac{22}{45} a^{5} - \frac{7}{15} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{135} a^{12} - \frac{1}{135} a^{10} + \frac{2}{45} a^{9} + \frac{2}{45} a^{8} - \frac{2}{15} a^{7} + \frac{4}{27} a^{6} - \frac{1}{15} a^{5} - \frac{10}{27} a^{4} + \frac{22}{45} a^{3} - \frac{1}{45} a^{2} - \frac{1}{3} a + \frac{4}{15}$, $\frac{1}{135} a^{13} - \frac{1}{135} a^{11} - \frac{1}{45} a^{9} - \frac{1}{15} a^{8} + \frac{11}{135} a^{7} + \frac{2}{15} a^{6} + \frac{22}{135} a^{5} - \frac{1}{15} a^{4} - \frac{19}{45} a^{3} + \frac{2}{5} a^{2} - \frac{2}{15} a - \frac{2}{5}$, $\frac{1}{405} a^{14} - \frac{1}{405} a^{13} - \frac{2}{405} a^{11} - \frac{4}{405} a^{10} + \frac{1}{27} a^{9} - \frac{19}{405} a^{8} + \frac{43}{405} a^{7} + \frac{2}{135} a^{6} - \frac{109}{405} a^{5} - \frac{34}{81} a^{4} + \frac{13}{45} a^{3} + \frac{13}{27} a^{2} - \frac{4}{9} a - \frac{8}{45}$, $\frac{1}{405} a^{15} - \frac{1}{405} a^{13} + \frac{1}{405} a^{12} + \frac{1}{135} a^{11} - \frac{1}{405} a^{10} - \frac{22}{405} a^{9} - \frac{4}{135} a^{8} + \frac{22}{405} a^{7} - \frac{16}{405} a^{6} + \frac{16}{45} a^{5} - \frac{167}{405} a^{4} + \frac{38}{135} a^{3} + \frac{29}{135} a^{2} - \frac{4}{45} a + \frac{4}{45}$, $\frac{1}{1215} a^{16} - \frac{1}{405} a^{13} - \frac{1}{135} a^{11} + \frac{13}{1215} a^{10} + \frac{1}{405} a^{9} - \frac{14}{405} a^{8} - \frac{47}{405} a^{7} - \frac{1}{27} a^{6} + \frac{59}{135} a^{5} + \frac{98}{243} a^{4} - \frac{43}{405} a^{3} - \frac{61}{405} a^{2} - \frac{7}{135} a - \frac{13}{27}$, $\frac{1}{3645} a^{17} + \frac{1}{3645} a^{16} + \frac{1}{1215} a^{14} + \frac{1}{405} a^{12} + \frac{2}{729} a^{11} + \frac{28}{3645} a^{10} - \frac{64}{1215} a^{9} - \frac{13}{135} a^{8} + \frac{28}{405} a^{7} + \frac{17}{405} a^{6} + \frac{781}{3645} a^{5} - \frac{1127}{3645} a^{4} + \frac{517}{1215} a^{3} + \frac{4}{243} a^{2} + \frac{26}{135} a + \frac{121}{405}$, $\frac{1}{4777424955} a^{18} + \frac{96869}{4777424955} a^{17} - \frac{482261}{4777424955} a^{16} - \frac{1888946}{1592474985} a^{15} + \frac{811822}{1592474985} a^{14} + \frac{101344}{106164999} a^{13} + \frac{15792697}{4777424955} a^{12} - \frac{37819807}{4777424955} a^{11} + \frac{1053839}{955484991} a^{10} + \frac{21563282}{1592474985} a^{9} + \frac{11181809}{106164999} a^{8} + \frac{87630932}{530824995} a^{7} + \frac{61469863}{4777424955} a^{6} + \frac{1772857733}{4777424955} a^{5} - \frac{341016674}{4777424955} a^{4} - \frac{233673227}{530824995} a^{3} + \frac{81242054}{1592474985} a^{2} - \frac{224937533}{530824995} a + \frac{163579606}{530824995}$, $\frac{1}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{19} - \frac{10755789632624937362024447365545676422923507623508050563485447394961901270174587424}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{18} - \frac{22303330778138971695785216839860585772278320675140197696573514707234134046205132834348246}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{17} - \frac{67879443401316939721026850524834666005100512422424981103611439564125541986186642146231822}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{16} - \frac{13822716369543995783931783784870095282563201831215952695286816171209583298143459140399648}{23885577724847408269439644704381007601747971483957987161466850984297272901561791368543055755} a^{15} + \frac{11622346652749198050918022050968192552576170236566711932924120133812072361805254476486154}{71656733174542224808318934113143022805243914451873961484400552952891818704685374105629167265} a^{14} - \frac{14087520205091521462728791003945428295705581596045573463561955494238688523328478286042961}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{13} + \frac{195207790195006198826584185716617153121517508486450777708836467500595516868517546296839968}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{12} + \frac{2276148917273492746941934678288552714702633814944617856263753445386529950614896422297755591}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{11} + \frac{898905287608914199262277061507879465618600945671073088317674985337100699438409722551655209}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{10} - \frac{1337818756126592426788285747102394065584271150598925267675650095188932925312409208068086083}{71656733174542224808318934113143022805243914451873961484400552952891818704685374105629167265} a^{9} + \frac{2745625329103237038251257627566170000088837951519716227179289612608999136763546466173474863}{23885577724847408269439644704381007601747971483957987161466850984297272901561791368543055755} a^{8} - \frac{14985294488838415817398132691633912579581033427372135201014716319542709000947929781815253881}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{7} + \frac{4528327288469260115386141150536335797389586206098111867552184521433741886959668489354575906}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{6} - \frac{29381754781894716432572535522373837871170105895964692993885505089867466471508054457138777326}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{5} + \frac{21902579888575368863442835734004691116876839511023663111264446996626395673694083602403198526}{214970199523626674424956802339429068415731743355621884453201658858675456114056122316887501795} a^{4} - \frac{16881594980309895836041999210556791387861511800126758200963449352221848740746082487896414461}{71656733174542224808318934113143022805243914451873961484400552952891818704685374105629167265} a^{3} - \frac{13666015983646766117700598066998090438129179387634446774505644999061675380647888833998671928}{71656733174542224808318934113143022805243914451873961484400552952891818704685374105629167265} a^{2} - \frac{1643499085544299099151120363942048531759414020504376648686928194192437408538992119150341838}{7961859241615802756479881568127002533915990494652662387155616994765757633853930456181018585} a - \frac{2098828017434126237613342704235420601626285150326608183164541905040603124531161016025034397}{4777115544969481653887928940876201520349594296791597432293370196859454580312358273708611151}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{4}\times C_{374412}$, which has order $383397888$ (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:  \( 3541438824.6395073 \) (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

\(\Q(\sqrt{205}) \), \(\Q(\sqrt{-246}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-30}, \sqrt{205})\), 5.5.2825761.1, 10.10.1023068544981128125.1, 10.0.2606819247971779313664.1, 10.0.198690491461263667200000.1

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 ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$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}$
41Data not computed