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Properties

Label 20.0.15327665644...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{32}\cdot 19^{10}$
Root discriminant $161.91$
Ramified primes $2, 5, 19$
Class number $14213265$ (GRH)
Class group $[11, 1292115]$ (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![4857308407, -3506883450, 3461090955, -1752920660, 1178825465, -531590874, 269657185, -99960640, 40600610, -13435700, 4543982, -1194250, 304360, -68930, 15720, -2108, 55, -10, 35, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 35*x^18 - 10*x^17 + 55*x^16 - 2108*x^15 + 15720*x^14 - 68930*x^13 + 304360*x^12 - 1194250*x^11 + 4543982*x^10 - 13435700*x^9 + 40600610*x^8 - 99960640*x^7 + 269657185*x^6 - 531590874*x^5 + 1178825465*x^4 - 1752920660*x^3 + 3461090955*x^2 - 3506883450*x + 4857308407)
 
gp: K = bnfinit(x^20 - 10*x^19 + 35*x^18 - 10*x^17 + 55*x^16 - 2108*x^15 + 15720*x^14 - 68930*x^13 + 304360*x^12 - 1194250*x^11 + 4543982*x^10 - 13435700*x^9 + 40600610*x^8 - 99960640*x^7 + 269657185*x^6 - 531590874*x^5 + 1178825465*x^4 - 1752920660*x^3 + 3461090955*x^2 - 3506883450*x + 4857308407, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 35 x^{18} - 10 x^{17} + 55 x^{16} - 2108 x^{15} + 15720 x^{14} - 68930 x^{13} + 304360 x^{12} - 1194250 x^{11} + 4543982 x^{10} - 13435700 x^{9} + 40600610 x^{8} - 99960640 x^{7} + 269657185 x^{6} - 531590874 x^{5} + 1178825465 x^{4} - 1752920660 x^{3} + 3461090955 x^{2} - 3506883450 x + 4857308407 \)

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:  \(153276656445025000000000000000000000000000000=2^{30}\cdot 5^{32}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $161.91$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3800=2^{3}\cdot 5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3800}(1,·)$, $\chi_{3800}(1861,·)$, $\chi_{3800}(2241,·)$, $\chi_{3800}(1481,·)$, $\chi_{3800}(1101,·)$, $\chi_{3800}(721,·)$, $\chi_{3800}(341,·)$, $\chi_{3800}(3001,·)$, $\chi_{3800}(3421,·)$, $\chi_{3800}(3041,·)$, $\chi_{3800}(2661,·)$, $\chi_{3800}(1521,·)$, $\chi_{3800}(2281,·)$, $\chi_{3800}(1901,·)$, $\chi_{3800}(2621,·)$, $\chi_{3800}(3761,·)$, $\chi_{3800}(1141,·)$, $\chi_{3800}(761,·)$, $\chi_{3800}(381,·)$, $\chi_{3800}(3381,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7} a^{7} - \frac{1}{7} a$, $\frac{1}{7} a^{8} - \frac{1}{7} a^{2}$, $\frac{1}{7} a^{9} - \frac{1}{7} a^{3}$, $\frac{1}{7} a^{10} - \frac{1}{7} a^{4}$, $\frac{1}{7} a^{11} - \frac{1}{7} a^{5}$, $\frac{1}{7} a^{12} - \frac{1}{7} a^{6}$, $\frac{1}{7} a^{13} - \frac{1}{7} a$, $\frac{1}{49} a^{14} - \frac{2}{49} a^{8} + \frac{1}{49} a^{2}$, $\frac{1}{1323} a^{15} + \frac{2}{441} a^{14} - \frac{13}{189} a^{13} - \frac{8}{189} a^{12} - \frac{8}{189} a^{11} - \frac{5}{189} a^{10} - \frac{16}{1323} a^{9} - \frac{4}{441} a^{8} - \frac{13}{189} a^{7} - \frac{76}{189} a^{6} - \frac{62}{189} a^{5} - \frac{8}{21} a^{4} - \frac{191}{441} a^{3} + \frac{349}{1323} a^{2} - \frac{58}{189} a - \frac{2}{27}$, $\frac{1}{9261} a^{16} - \frac{1}{3087} a^{15} + \frac{17}{9261} a^{14} + \frac{82}{1323} a^{13} + \frac{10}{1323} a^{12} - \frac{68}{1323} a^{11} - \frac{646}{9261} a^{10} - \frac{82}{3087} a^{9} + \frac{449}{9261} a^{8} - \frac{94}{1323} a^{7} + \frac{298}{1323} a^{6} + \frac{9}{49} a^{5} + \frac{754}{3087} a^{4} - \frac{731}{9261} a^{3} - \frac{1495}{9261} a^{2} + \frac{481}{1323} a + \frac{8}{21}$, $\frac{1}{9261} a^{17} + \frac{1}{9261} a^{15} + \frac{16}{9261} a^{14} - \frac{31}{1323} a^{13} + \frac{2}{147} a^{12} - \frac{359}{9261} a^{11} - \frac{88}{1323} a^{10} - \frac{59}{3087} a^{9} + \frac{584}{9261} a^{8} - \frac{82}{1323} a^{7} + \frac{346}{1323} a^{6} - \frac{181}{9261} a^{5} - \frac{143}{1323} a^{4} + \frac{323}{9261} a^{3} - \frac{935}{3087} a^{2} + \frac{274}{1323} a + \frac{41}{189}$, $\frac{1}{7242141265991665173} a^{18} + \frac{80974233407444}{2414047088663888391} a^{17} - \frac{214052739661493}{7242141265991665173} a^{16} - \frac{40717257270440}{804682362887962797} a^{15} - \frac{56999531430464437}{7242141265991665173} a^{14} - \frac{42741310860919772}{1034591609427380739} a^{13} + \frac{12926734583305486}{804682362887962797} a^{12} - \frac{97878338266902992}{7242141265991665173} a^{11} + \frac{206708644711326719}{7242141265991665173} a^{10} + \frac{10165773048802186}{804682362887962797} a^{9} - \frac{501767635678780717}{7242141265991665173} a^{8} + \frac{10584062363081327}{1034591609427380739} a^{7} + \frac{159548621979609496}{804682362887962797} a^{6} + \frac{126068119098479078}{268227454295987599} a^{5} + \frac{2155339019695164935}{7242141265991665173} a^{4} - \frac{1203313772249287628}{2414047088663888391} a^{3} - \frac{282895876260485864}{804682362887962797} a^{2} + \frac{1165553298968719}{8020089995561091} a + \frac{63468165116367884}{147798801346768677}$, $\frac{1}{729109845832936637890483511897977924874423139} a^{19} - \frac{9054029841629768386848298}{729109845832936637890483511897977924874423139} a^{18} + \frac{24560559682464300452775375594234402929359}{729109845832936637890483511897977924874423139} a^{17} + \frac{2973366585279555620271459104896938058481}{729109845832936637890483511897977924874423139} a^{16} + \frac{228086712265554262893425159006441160203230}{729109845832936637890483511897977924874423139} a^{15} - \frac{376613744390564609503533878548427565075250}{104158549404705233984354787413996846410631877} a^{14} - \frac{5757189142528418105114276827652266322998370}{243036615277645545963494503965992641624807713} a^{13} + \frac{2287170328396755667887078953908266000608612}{81012205092548515321164834655330880541602571} a^{12} + \frac{11679380624130612245956825248191807735074138}{243036615277645545963494503965992641624807713} a^{11} - \frac{1564757681853798385499681574718938089905081}{243036615277645545963494503965992641624807713} a^{10} + \frac{6190805830432034312991202023061301535515402}{243036615277645545963494503965992641624807713} a^{9} - \frac{846377438720407045173848902255969114718426}{34719516468235077994784929137998948803543959} a^{8} - \frac{6712554416982640966877323267050596946350233}{729109845832936637890483511897977924874423139} a^{7} + \frac{216183523740756356825916584408520319404074608}{729109845832936637890483511897977924874423139} a^{6} - \frac{243446871455491731309417217176373192462774829}{729109845832936637890483511897977924874423139} a^{5} - \frac{6279898124933031706139710685655042121045541}{16956042926347363671871709579022742438940073} a^{4} - \frac{31965041743194802279116974313547420651788145}{81012205092548515321164834655330880541602571} a^{3} - \frac{39087694728686727922277461695448083586512556}{104158549404705233984354787413996846410631877} a^{2} - \frac{26325100577373293293660677164126939101988}{708561560576226081526223043632631608235591} a - \frac{296988425765424920653227908879352589274230}{708561560576226081526223043632631608235591}$

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

Class group and class number

$C_{11}\times C_{1292115}$, which has order $14213265$ (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:  \( 42294001.73672045 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-38}) \), \(\Q(\sqrt{2}, \sqrt{-19})\), 5.5.390625.1, 10.10.5000000000000000.1, 10.0.377822723388671875.3, 10.0.12380495000000000000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
5.10.16.7$x^{10} + 40 x^{9} + 10 x^{8} + 70 x^{7} + 15 x^{6} + 95 x^{5} + 5 x^{4} + 80 x^{3} + 5 x^{2} + 90 x + 7$$5$$2$$16$$C_{10}$$[2]^{2}$
19Data not computed