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Properties

Label 20.0.34235216575...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{27}\cdot 11^{16}$
Root discriminant $59.80$
Ramified primes $5, 11$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (GRH)
Galois group $C_5:C_4$ (as 20T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![105820496, 310166920, 382849660, 287303870, 177521795, 98052526, 39422120, 12523855, 5882110, 2491990, 313441, -52935, 44790, 26665, -1355, -1859, 220, 120, -20, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 20*x^18 + 120*x^17 + 220*x^16 - 1859*x^15 - 1355*x^14 + 26665*x^13 + 44790*x^12 - 52935*x^11 + 313441*x^10 + 2491990*x^9 + 5882110*x^8 + 12523855*x^7 + 39422120*x^6 + 98052526*x^5 + 177521795*x^4 + 287303870*x^3 + 382849660*x^2 + 310166920*x + 105820496)
 
gp: K = bnfinit(x^20 - 5*x^19 - 20*x^18 + 120*x^17 + 220*x^16 - 1859*x^15 - 1355*x^14 + 26665*x^13 + 44790*x^12 - 52935*x^11 + 313441*x^10 + 2491990*x^9 + 5882110*x^8 + 12523855*x^7 + 39422120*x^6 + 98052526*x^5 + 177521795*x^4 + 287303870*x^3 + 382849660*x^2 + 310166920*x + 105820496, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 20 x^{18} + 120 x^{17} + 220 x^{16} - 1859 x^{15} - 1355 x^{14} + 26665 x^{13} + 44790 x^{12} - 52935 x^{11} + 313441 x^{10} + 2491990 x^{9} + 5882110 x^{8} + 12523855 x^{7} + 39422120 x^{6} + 98052526 x^{5} + 177521795 x^{4} + 287303870 x^{3} + 382849660 x^{2} + 310166920 x + 105820496 \)

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:  \(342352165755422122776508331298828125=5^{27}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.80$
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:  $5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{14} a^{17} - \frac{1}{7} a^{16} - \frac{1}{14} a^{15} - \frac{1}{7} a^{14} - \frac{1}{7} a^{13} + \frac{1}{7} a^{12} - \frac{1}{14} a^{11} - \frac{1}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{14} a^{8} - \frac{5}{14} a^{7} + \frac{1}{7} a^{6} - \frac{5}{14} a^{5} - \frac{2}{7} a^{4} + \frac{1}{14} a^{3} - \frac{3}{7} a^{2} - \frac{1}{14} a + \frac{1}{7}$, $\frac{1}{196} a^{18} - \frac{3}{196} a^{17} + \frac{11}{98} a^{16} + \frac{5}{49} a^{15} + \frac{3}{14} a^{14} - \frac{31}{196} a^{13} + \frac{11}{196} a^{12} + \frac{13}{196} a^{11} - \frac{15}{98} a^{10} - \frac{39}{196} a^{9} - \frac{11}{196} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{83}{196} a^{5} + \frac{10}{49} a^{4} - \frac{3}{7} a^{3} + \frac{75}{196} a^{2} + \frac{13}{49} a + \frac{24}{49}$, $\frac{1}{574819044391008693242641621845540404377867594383790476738249781160356024} a^{19} + \frac{948069576790597331750665839993952141390830020792416302738751868116873}{574819044391008693242641621845540404377867594383790476738249781160356024} a^{18} - \frac{3040183887472229610783354773552794895548769126689016801297743104208483}{287409522195504346621320810922770202188933797191895238369124890580178012} a^{17} - \frac{17964066902773231147470618933730305787586172091733846811660067920199141}{143704761097752173310660405461385101094466898595947619184562445290089006} a^{16} - \frac{2529948938720930254583518731767632107599911392817723093800855866069429}{143704761097752173310660405461385101094466898595947619184562445290089006} a^{15} - \frac{82842499913894318173423076722995910704400655026670192253035537351921427}{574819044391008693242641621845540404377867594383790476738249781160356024} a^{14} - \frac{1455823153646690393832547543878916689111207250890135051493831340510427}{82117006341572670463234517406505772053981084911970068105464254451479432} a^{13} + \frac{184411688594273403656620133398249108605488717221251528082296453811497}{765404852717721295928950228822290818079717169618895441728694781838024} a^{12} - \frac{25961820658927434486500123996471645007126080097455194115642445025377735}{143704761097752173310660405461385101094466898595947619184562445290089006} a^{11} - \frac{272294955829189062251613952855920872497689464720788045807880537173096411}{574819044391008693242641621845540404377867594383790476738249781160356024} a^{10} - \frac{17943357247183020037060801232387324826192516580266736265316974398621983}{82117006341572670463234517406505772053981084911970068105464254451479432} a^{9} - \frac{3948603520116720474481066126035308754581037892332601766835731590771423}{143704761097752173310660405461385101094466898595947619184562445290089006} a^{8} - \frac{13532116233836857916169219644668662878774100238302043997903639843980225}{41058503170786335231617258703252886026990542455985034052732127225739716} a^{7} + \frac{230534918160339211977821258659737777346898383212295080003979231896262027}{574819044391008693242641621845540404377867594383790476738249781160356024} a^{6} + \frac{67380810980601158132862284608298497620648246822066851939252863272384081}{287409522195504346621320810922770202188933797191895238369124890580178012} a^{5} - \frac{3093233140570211429664047003815391891414620043368675184154694504535661}{287409522195504346621320810922770202188933797191895238369124890580178012} a^{4} - \frac{179067273024336806387338209731096141335996684544112367197289322649202937}{574819044391008693242641621845540404377867594383790476738249781160356024} a^{3} + \frac{43467691087914490305911635587607675466035916269710331543044788428521543}{143704761097752173310660405461385101094466898595947619184562445290089006} a^{2} + \frac{33467324802913258016547637847287817583668652614216360453413834449380315}{143704761097752173310660405461385101094466898595947619184562445290089006} a + \frac{11031307861907477604266099902613599391202908892164964999696459668854544}{71852380548876086655330202730692550547233449297973809592281222645044503}$

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

Class group and class number

$C_{5}\times C_{5}\times C_{5}$, which has order $125$ (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:  \( -\frac{39979359349790292388674222976561096268389319314750986689}{23476683183766082036910726883476721050991980876869183835258952} a^{19} + \frac{239973864625540499345092666539083823864430470721903070775}{23476683183766082036910726883476721050991980876869183835258952} a^{18} + \frac{285740142846844192172946915316858747597259241704266909081}{11738341591883041018455363441738360525495990438434591917629476} a^{17} - \frac{195385305312954837752834070661864436595996150016955975523}{838452970848788644175383102981311466106856459888185136973534} a^{16} - \frac{410475602306935659241204716963988875326155710318177066608}{2934585397970760254613840860434590131373997609608647979407369} a^{15} + \frac{79268357348193688614210572174659143576776121296431525699683}{23476683183766082036910726883476721050991980876869183835258952} a^{14} - \frac{27654390491331475886015615279356338319158377185621392237663}{23476683183766082036910726883476721050991980876869183835258952} a^{13} - \frac{1409263412859076736670943320755753199204250386376345376073}{31260563493696514030506959898104821639137125002488926544952} a^{12} - \frac{84994217980687087041069183378352290084333559220248222536903}{2934585397970760254613840860434590131373997609608647979407369} a^{11} + \frac{3028085521082289290041053362828847319589801845262353920451695}{23476683183766082036910726883476721050991980876869183835258952} a^{10} - \frac{15829700475015083622534925699739287289707469103135961296649259}{23476683183766082036910726883476721050991980876869183835258952} a^{9} - \frac{10525213993142681678421979642872480506378635443055733709156568}{2934585397970760254613840860434590131373997609608647979407369} a^{8} - \frac{10351445835405289208391950693244412817562531246040641115647815}{1676905941697577288350766205962622932213712919776370273947068} a^{7} - \frac{340993641586672122920247937931074407518785680076711524329259475}{23476683183766082036910726883476721050991980876869183835258952} a^{6} - \frac{609582774989700974569640390949625192156544066555287325254664933}{11738341591883041018455363441738360525495990438434591917629476} a^{5} - \frac{1312677213921986738711157063262312556767973961230873114280152085}{11738341591883041018455363441738360525495990438434591917629476} a^{4} - \frac{4238137811685316496551798932969682523237317592114637497033265703}{23476683183766082036910726883476721050991980876869183835258952} a^{3} - \frac{861211788354217817524629257503857260766732491848090118358533673}{2934585397970760254613840860434590131373997609608647979407369} a^{2} - \frac{976479208807171649231376249839892320068321025671986825132232974}{2934585397970760254613840860434590131373997609608647979407369} a - \frac{458206037968230307377288715303702855285161528658957744978106074}{2934585397970760254613840860434590131373997609608647979407369} \) (order $10$)
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:  \( 143712739.876 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5:C_4$ (as 20T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 8 conjugacy class representatives for $C_5:C_4$
Character table for $C_5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.228765625.1 x5, 10.10.261668555908203125.1

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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
5Data not computed
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$