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Properties

Label 20.0.25422620064...8125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 11^{12}\cdot 61^{12}$
Root discriminant $166.06$
Ramified primes $5, 11, 61$
Class number $125$ (GRH)
Class group $[5, 5, 5]$ (GRH)
Galois group $C_5\times F_5$ (as 20T29)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86798990281751, 29747851180432, -8270841308686, -7823618530310, -502397221205, 717838125680, 214119303238, -16433231696, -17208973972, -2282213678, 569808055, 188733886, 7151967, -5128108, -735310, 38288, 17039, 374, -167, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 - 167*x^18 + 374*x^17 + 17039*x^16 + 38288*x^15 - 735310*x^14 - 5128108*x^13 + 7151967*x^12 + 188733886*x^11 + 569808055*x^10 - 2282213678*x^9 - 17208973972*x^8 - 16433231696*x^7 + 214119303238*x^6 + 717838125680*x^5 - 502397221205*x^4 - 7823618530310*x^3 - 8270841308686*x^2 + 29747851180432*x + 86798990281751)
 
gp: K = bnfinit(x^20 - 8*x^19 - 167*x^18 + 374*x^17 + 17039*x^16 + 38288*x^15 - 735310*x^14 - 5128108*x^13 + 7151967*x^12 + 188733886*x^11 + 569808055*x^10 - 2282213678*x^9 - 17208973972*x^8 - 16433231696*x^7 + 214119303238*x^6 + 717838125680*x^5 - 502397221205*x^4 - 7823618530310*x^3 - 8270841308686*x^2 + 29747851180432*x + 86798990281751, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} - 167 x^{18} + 374 x^{17} + 17039 x^{16} + 38288 x^{15} - 735310 x^{14} - 5128108 x^{13} + 7151967 x^{12} + 188733886 x^{11} + 569808055 x^{10} - 2282213678 x^{9} - 17208973972 x^{8} - 16433231696 x^{7} + 214119303238 x^{6} + 717838125680 x^{5} - 502397221205 x^{4} - 7823618530310 x^{3} - 8270841308686 x^{2} + 29747851180432 x + 86798990281751 \)

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:  \(254226200642572774598633256129316436767578125=5^{15}\cdot 11^{12}\cdot 61^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $166.06$
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:  $5, 11, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{3}$, $\frac{1}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{19} - \frac{6175124551240117509531151107743978853069026521502174489377813916347509422148596335249557378280156064479642207087558434627682}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{18} + \frac{35788319317532276524031008949110923900728016394173003663203889709254939310674023964155472778307360045073996426203455646033392}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{17} - \frac{30158849359133076408413351545265978074342129999763532835620709399527719841651030687935774747095435906616395423605694433468539}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{16} + \frac{10418029184978795445232731444504698737589840581586233458592836225627153803292722693024645655827754322820442914101930206848687}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{15} - \frac{18273971226845240226930657831718073927284700772467042773042990553260417103338844118644860011937249656688141155089109964062925}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{14} + \frac{33748199988131645776537981953854520803428455077388381391801161348260260736335945607289322761095979732737452267518154865270773}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{13} + \frac{51725670828131450995922836478296684244795820826098101501016877497917667602576495972640002343869517520654872155482670865481939}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{12} - \frac{34198128317612270815658516350075462321743447080110635735464374733806527838895017201462239783749508983038852488991932379055343}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{11} + \frac{65296973249266449638408251760062839537666237694257170062722499380263889681413219219225917220491932519439408914763508785835141}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{10} - \frac{25317375560102354621867169930418460661954383728299365553908655283142570246704316706544755358953965632771457364376439351977327}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{9} + \frac{42916373640880589629950822217428002994910749079693440727242043622078632844795909904160858053082350252835669183846382294405503}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{8} + \frac{90376010123933523983764076844718259529208603124151704938766026945951367348301969626559055417204001638803781088776638408374413}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{7} + \frac{4190038702307526918497198230026112418614647416058418427124647745645573493172065005398850540939418831526564181411453996372323}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{6} - \frac{25877703683811794801892970852459742347653890319898735097382819316061690947774058143035686145039764338510689999404478752363163}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{5} + \frac{139660938880921219539170373167284867100930823403718676138885334092872728451893943318085067739585200088562144933109834757293963}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682} a^{4} + \frac{68249411314848803284408615800444255194668489544693061137583388560251836039667511820128914530445395267679149572895408929892808}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{3} + \frac{41842839990630426329126549339095151848435140143337409671890430561330176010860305550970594289183495660427399512861362944522423}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a^{2} + \frac{9021817124879403947791039038747643985799915916564681855455501127412439647344742386242983497154460509044599221667960897625748}{143767218734997073438411518306895705185334750860155373466113066251233916643090144557688793886936719732335095671381618669963841} a - \frac{40554911708598492509408911349852593215128951764646679995479058328400386642444535383000233043768542579705696369362216686515343}{287534437469994146876823036613791410370669501720310746932226132502467833286180289115377587773873439464670191342763237339927682}$

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{17130066970667479956881483611385882519350697912778613706737847134934523437875948966167642614855}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{19} + \frac{1134574159757406296595984690664691392874156859218460024862326153305736732846530000934569041980217}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{18} - \frac{3192245000100169646731652006569985749244076275193587666628037091618721217410376541086695365681153}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{17} - \frac{50602313600623285697514146978908794759223262201262993542801922979489522766012384199264234493732236}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{16} + \frac{228614617389781942132239594272468201169496899837386301886443183837106250606156580630438076789097379}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{15} + \frac{4739931964919707860355335260482036984380640528991662648769837237735200208582461779834008787715235163}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{14} - \frac{8910806254531803559559218503374031029611608138328158727058739096818088492929005887413283084636042878}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{13} - \frac{446922780387038626479481661680002340302288151720456314630300161196144069849131934986969112447339328367}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{12} - \frac{1093827118736678984142239314747441881599790453145497770422801823575106554820014452853058395695899295445}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{11} + \frac{7548348018367204528952013183132110100871499743381314239868928945859194825024322143760230874893210754626}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{10} + \frac{72269676215920760388511372685649181639502596994959327422095663207344425143984126600430033007807280303125}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{9} - \frac{54741343351850738753759121904415269911792173526824063669387384147195910814815232530727489818691018957037}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{8} - \frac{2268435184311332857484837855808341610678296134521083221141403687599201968289482677947903679998113734588979}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{7} - \frac{2893048665191398721955544867323345092705656810247221809283918974388126823446076479294341363921346187687691}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{6} + \frac{19011532049087689296096304215054766739057354444648381754773868956816201528076073549976945260699528214109883}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{5} + \frac{119969150372408354631450746262859956223157036391026544214838399306836093207666868998279882864904942679365375}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{4} - \frac{67481129051731087457769051440172257754204957925310664517133339960680978294103374475767199014553690257203113}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} a^{3} - \frac{445927980625320542943347994508394659549704676777318509123674651341814512789713562658170540903746961795176829}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a^{2} - \frac{782396248910658065949839450855905985717181997766402310589984555384646793285725095560190555158810982332925743}{959101289512506796186052690713451004635399055896814555698229889083474020052061735925541759218915396303922851} a + \frac{5619487591377328101803179839413674682361650679238564768215630018193469096726341455456941375533859631132229545}{1918202579025013592372105381426902009270798111793629111396459778166948040104123471851083518437830792607845702} \) (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:  \( 3970645939642.0083 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times F_5$ (as 20T29):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 100
The 25 conjugacy class representatives for $C_5\times F_5$
Character table for $C_5\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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

Sibling fields

Degree 25 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ $20$ R $20$ R $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{5}$ $20$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$11.5.4.5$x^{5} - 99$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.0.1$x^{5} + x^{2} - x + 5$$1$$5$$0$$C_5$$[\ ]^{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.1$x^{5} + 297$$5$$1$$4$$C_5$$[\ ]_{5}$
$61$61.5.0.1$x^{5} - x + 6$$1$$5$$0$$C_5$$[\ ]^{5}$
61.5.4.1$x^{5} - 61$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.3$x^{5} - 244$$5$$1$$4$$C_5$$[\ ]_{5}$
61.5.4.4$x^{5} + 488$$5$$1$$4$$C_5$$[\ ]_{5}$