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Properties

Label 20.0.47096078384...4213.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{9}\cdot 29^{10}$
Root discriminant $68.18$
Ramified primes $7, 11, 29$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![79154483718199, 21873265752274, 3628004636611, -785792800744, 202954019677, -17686730229, -911324443, 1783530285, 673956560, -122674031, -10904667, -2397999, 1049707, 32718, 50894, -8040, 346, -96, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 - 96*x^17 + 346*x^16 - 8040*x^15 + 50894*x^14 + 32718*x^13 + 1049707*x^12 - 2397999*x^11 - 10904667*x^10 - 122674031*x^9 + 673956560*x^8 + 1783530285*x^7 - 911324443*x^6 - 17686730229*x^5 + 202954019677*x^4 - 785792800744*x^3 + 3628004636611*x^2 + 21873265752274*x + 79154483718199)
 
gp: K = bnfinit(x^20 - 2*x^19 - 96*x^17 + 346*x^16 - 8040*x^15 + 50894*x^14 + 32718*x^13 + 1049707*x^12 - 2397999*x^11 - 10904667*x^10 - 122674031*x^9 + 673956560*x^8 + 1783530285*x^7 - 911324443*x^6 - 17686730229*x^5 + 202954019677*x^4 - 785792800744*x^3 + 3628004636611*x^2 + 21873265752274*x + 79154483718199, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} - 96 x^{17} + 346 x^{16} - 8040 x^{15} + 50894 x^{14} + 32718 x^{13} + 1049707 x^{12} - 2397999 x^{11} - 10904667 x^{10} - 122674031 x^{9} + 673956560 x^{8} + 1783530285 x^{7} - 911324443 x^{6} - 17686730229 x^{5} + 202954019677 x^{4} - 785792800744 x^{3} + 3628004636611 x^{2} + 21873265752274 x + 79154483718199 \)

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:  \(4709607838486813481044153118961614213=7^{15}\cdot 11^{9}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $68.18$
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:  $7, 11, 29$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{19} + \frac{6497436279113146548503162196482416242566473427103788827675493989317636346927853390803145843749177887863055428341637351567786273059}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{18} - \frac{10496682858236389677488051953232435494991511563695859464802510370716171182193050278188126678177044792184996874394239742920980331451}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{17} + \frac{1544962353468861028993826552188162870536755419911376915281061830246622529011399260370704675650250422871898968388239764815420919115}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{16} - \frac{16583026167207409061361891441948274772136441933292332832416845493455011242981441565564881229378873755550614532152461359133015911518}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{15} - \frac{30919366571787102470036495229460697823793648698159866678703583831441637736181365663240252603847654565731096102491147429452737939043}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{14} + \frac{13644750612499519843133355877043306460930934878665161930187654273543732776026719112281400343055367650319821429027530541744996645073}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{13} + \frac{841675399699977433348922886707675073358670585039486376561205352815723435276556483332689098428827999119084660132427989405811688155}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{12} - \frac{3339357547147905707907813512376971658814754038843675302237286229960982793653446828097820189075209447100733132062912489063568749167}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{11} - \frac{18727734422821636496975945620353655083917251115373453493129014032523965778090191330093491184047204205413623210267671982890982646771}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{10} - \frac{23880834436965250580386430067779724812142009577142739004804112004022248901940883770124660251580193198409444241768917784032594043738}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{9} - \frac{26588829165862798549262104373303923049203477474646737968894902512778764534125056204261891168621983215160251070865156258772007779336}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{8} + \frac{21576895577638926126315488481398871081427545508323039125387951763722244678017403235384655450370500582438422852225376334026211736125}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{7} + \frac{1975452068446800782296609773237686293156991306209815906077957020861130313703590037673499492313815164337700474922532214722168236073}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{6} + \frac{16230975694517291867128282978332331775345154132437830903192714068967759517016400837810736269244712713882173513985123105873039900009}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{5} - \frac{3840475594558383929320545519624293114795845479909551138932553300052684741154792173468631069389133680517978290411913097220769990873}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{4} + \frac{11197521855774211365186745162138783877921094877200134434173061067595188167856516530714495893840118292778975870704041539877127816631}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{3} + \frac{27368479422575938648001920814235618664907111934128233823531252750323019154843989323751543210266166388750184888358906532270587940290}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a^{2} + \frac{14332056571729418752848449482513234267612651653399385004840265968146096701964164700840295112529915945258870051073930992022649901036}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409} a + \frac{20880425640683693566194006835348809047773742734599373037730740870121558081688799345647751463364554235714425970018206703250936300411}{62718554277129854059129434435774281960897421007759091725776056425138942515612804839657607752532030364915942104473223301519344163409}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.3173093.1, 10.0.246071287.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 siblings: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
7Data not computed
$11$11.10.9.5$x^{10} - 8019$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
$29$29.10.5.1$x^{10} - 1682 x^{6} + 707281 x^{2} - 2481849029$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
29.10.5.2$x^{10} - 707281 x^{2} + 225622639$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$