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Properties

Label 20.0.75263294618...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 13^{15}$
Root discriminant $311.77$
Ramified primes $2, 5, 11, 13$
Class number $742073296$ (GRH)
Class group $[2, 2, 185518324]$ (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![7031931591101, -1917948259382, 4136129114757, -997073174094, 1203330073916, -262142242452, 224203197309, -40062680052, 27675900374, -4536732452, 2472180640, -259868396, 115326734, -7815380, 2923384, -128764, 41560, -1114, 315, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 315*x^18 - 1114*x^17 + 41560*x^16 - 128764*x^15 + 2923384*x^14 - 7815380*x^13 + 115326734*x^12 - 259868396*x^11 + 2472180640*x^10 - 4536732452*x^9 + 27675900374*x^8 - 40062680052*x^7 + 224203197309*x^6 - 262142242452*x^5 + 1203330073916*x^4 - 997073174094*x^3 + 4136129114757*x^2 - 1917948259382*x + 7031931591101)
 
gp: K = bnfinit(x^20 - 4*x^19 + 315*x^18 - 1114*x^17 + 41560*x^16 - 128764*x^15 + 2923384*x^14 - 7815380*x^13 + 115326734*x^12 - 259868396*x^11 + 2472180640*x^10 - 4536732452*x^9 + 27675900374*x^8 - 40062680052*x^7 + 224203197309*x^6 - 262142242452*x^5 + 1203330073916*x^4 - 997073174094*x^3 + 4136129114757*x^2 - 1917948259382*x + 7031931591101, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 315 x^{18} - 1114 x^{17} + 41560 x^{16} - 128764 x^{15} + 2923384 x^{14} - 7815380 x^{13} + 115326734 x^{12} - 259868396 x^{11} + 2472180640 x^{10} - 4536732452 x^{9} + 27675900374 x^{8} - 40062680052 x^{7} + 224203197309 x^{6} - 262142242452 x^{5} + 1203330073916 x^{4} - 997073174094 x^{3} + 4136129114757 x^{2} - 1917948259382 x + 7031931591101 \)

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:  \(75263294618341622670336367123708064000000000000000=2^{20}\cdot 5^{15}\cdot 11^{16}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $311.77$
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:  $2, 5, 11, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2860=2^{2}\cdot 5\cdot 11\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2860}(1,·)$, $\chi_{2860}(1347,·)$, $\chi_{2860}(389,·)$, $\chi_{2860}(1607,·)$, $\chi_{2860}(521,·)$, $\chi_{2860}(2127,·)$, $\chi_{2860}(2729,·)$, $\chi_{2860}(1169,·)$, $\chi_{2860}(1301,·)$, $\chi_{2860}(603,·)$, $\chi_{2860}(863,·)$, $\chi_{2860}(2209,·)$, $\chi_{2860}(1123,·)$, $\chi_{2860}(2469,·)$, $\chi_{2860}(2601,·)$, $\chi_{2860}(1643,·)$, $\chi_{2860}(47,·)$, $\chi_{2860}(2423,·)$, $\chi_{2860}(2341,·)$, $\chi_{2860}(1087,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{8} - \frac{1}{7} a^{7} + \frac{2}{7} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{2}{7} a^{2} + \frac{2}{7} a - \frac{2}{7}$, $\frac{1}{7} a^{11} + \frac{1}{7} a^{9} + \frac{2}{7} a^{8} - \frac{2}{7} a^{7} + \frac{2}{7} a^{6} - \frac{2}{7} a^{5} - \frac{2}{7} a^{4} + \frac{1}{7} a^{3} - \frac{1}{7} a^{2} + \frac{2}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{12} - \frac{3}{7} a^{9} + \frac{3}{7} a^{7} - \frac{2}{7} a^{6} + \frac{3}{7} a^{5} + \frac{3}{7} a^{3} + \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{13} + \frac{1}{7} a^{9} - \frac{3}{7} a^{8} + \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{1}{7} a^{5} - \frac{1}{7} a^{4} + \frac{2}{7} a^{3} + \frac{1}{7} a + \frac{1}{7}$, $\frac{1}{7} a^{14} - \frac{1}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} + \frac{1}{7} a^{4} - \frac{3}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{2}{7}$, $\frac{1}{14} a^{15} - \frac{1}{14} a^{14} - \frac{1}{14} a^{12} - \frac{1}{14} a^{11} - \frac{1}{14} a^{10} - \frac{1}{2} a^{9} + \frac{5}{14} a^{8} - \frac{3}{7} a^{7} + \frac{5}{14} a^{6} - \frac{2}{7} a^{5} - \frac{1}{7} a^{4} - \frac{1}{7} a^{3} + \frac{1}{14} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{16} - \frac{1}{14} a^{14} - \frac{1}{14} a^{13} + \frac{3}{7} a^{9} + \frac{1}{14} a^{8} - \frac{1}{2} a^{7} + \frac{1}{14} a^{6} - \frac{1}{7} a^{5} + \frac{3}{14} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a + \frac{1}{14}$, $\frac{1}{14} a^{17} - \frac{1}{14} a^{12} - \frac{1}{14} a^{11} - \frac{1}{14} a^{10} + \frac{2}{7} a^{9} + \frac{2}{7} a^{8} - \frac{5}{14} a^{7} + \frac{1}{14} a^{6} + \frac{3}{7} a^{5} - \frac{3}{14} a^{4} - \frac{2}{7} a^{3} - \frac{5}{14} a^{2} - \frac{3}{7} a - \frac{5}{14}$, $\frac{1}{7761624078151162505184000199023141614} a^{18} + \frac{13945965523677733900555334780145521}{3880812039075581252592000099511570807} a^{17} - \frac{62022761124205249250531187508034617}{3880812039075581252592000099511570807} a^{16} - \frac{4762120118582176817178034385669827}{7761624078151162505184000199023141614} a^{15} - \frac{501618895929729351963443650892586831}{7761624078151162505184000199023141614} a^{14} - \frac{341666376686485758644660307276369595}{7761624078151162505184000199023141614} a^{13} - \frac{164052621738676325473997773575602487}{3880812039075581252592000099511570807} a^{12} + \frac{207705559604679594791270401521006426}{3880812039075581252592000099511570807} a^{11} + \frac{119994535638968590084416898895386407}{7761624078151162505184000199023141614} a^{10} + \frac{987827204881206610824065172326446113}{7761624078151162505184000199023141614} a^{9} + \frac{1098826297290816204466053656146160964}{3880812039075581252592000099511570807} a^{8} - \frac{356850497897939438988435727671724957}{7761624078151162505184000199023141614} a^{7} + \frac{3752552664458886318584582074366500295}{7761624078151162505184000199023141614} a^{6} + \frac{267322975351844924664668639425752277}{1108803439735880357883428599860448802} a^{5} + \frac{493826993831011264876058338559653253}{3880812039075581252592000099511570807} a^{4} + \frac{197928797770772625415689158969111663}{1108803439735880357883428599860448802} a^{3} + \frac{2236718095811282068076298489217193225}{7761624078151162505184000199023141614} a^{2} - \frac{416503288501680901097975058112176681}{3880812039075581252592000099511570807} a + \frac{8135025775598248470013615366570429}{19550690373176731751093199493761062}$, $\frac{1}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{19} + \frac{5314270289097584234801108605242402223}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{18} + \frac{2370873063204418902028120388118159742272214930050027164981516998748245729}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{17} - \frac{1353681644424691756338596792727740870766720172956971586766416439379331620}{60467549259229653434605279278168565360269275337996791044658051461518518827} a^{16} + \frac{1041220008162079868888535723514463913054126557508396707682163910531720249}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{15} + \frac{2630532753060739456980941931932906634277505445194778311962637234464129504}{60467549259229653434605279278168565360269275337996791044658051461518518827} a^{14} - \frac{1393183735138404491872341407149955552873566923523923141054375554921201408}{60467549259229653434605279278168565360269275337996791044658051461518518827} a^{13} + \frac{2005835847093621052769327206906590042196297893547945670871839579785686618}{60467549259229653434605279278168565360269275337996791044658051461518518827} a^{12} - \frac{2143585954762358598841492203116349072392118211069232444210787646897121441}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{11} - \frac{522267414041573841206636831720463257874544885780815063494859362770171332}{8638221322747093347800754182595509337181325048285255863522578780216931261} a^{10} - \frac{29315549867122043300353379760986259334412404921227319843686293646325979531}{60467549259229653434605279278168565360269275337996791044658051461518518827} a^{9} + \frac{10441136719326864682859238047263078606762869273847642676889147222897948167}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{8} - \frac{6450640784607988310651633867085170830028413985221944286671596832551922648}{60467549259229653434605279278168565360269275337996791044658051461518518827} a^{7} - \frac{511990982945753095746299640072196675244032706490617946949447561998252299}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{6} - \frac{28354170834454459047451266378950740090580174841502027878843415309822573499}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{5} + \frac{1757909053398580348696328389088112541528791603628411186149958952178772013}{8638221322747093347800754182595509337181325048285255863522578780216931261} a^{4} + \frac{3846110877952009187332775013393201424583593571210104690963192525006950777}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{3} + \frac{3597846338488005008990133515564072566163302148932102075938996641601470839}{120935098518459306869210558556337130720538550675993582089316102923037037654} a^{2} - \frac{14418522030891949782338021442965830361106213817184412154005349082795810809}{60467549259229653434605279278168565360269275337996791044658051461518518827} a - \frac{31728125451131081893344331474807952449096611934672295899546294305460475}{152311207202089807140063675763648779245010769113342042933647484789719191}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{185518324}$, which has order $742073296$ (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:  \( 40471423.40714512 \) (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

\(\Q(\sqrt{65}) \), 4.0.4394000.1, \(\Q(\zeta_{11})^+\), 10.10.248718600009790625.1

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 $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R R $20$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ $20$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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$2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
2.10.10.11$x^{10} - x^{8} + 3 x^{6} + x^{2} - 3$$2$$5$$10$$C_{10}$$[2]^{5}$
5Data not computed
11Data not computed
13Data not computed