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Properties

Label 20.4.19867324198...0625.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{10}\cdot 4588681^{5}$
Root discriminant $103.49$
Ramified primes $5, 4588681$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group 20T1022

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125053657678981, 66656193013587, 30961063038531, 11858163795028, 3108087294944, 650613452731, 89634505875, 10583460950, 1019228952, -319740577, -43052462, -11246243, -5912224, -215365, -126061, -15417, 2959, -455, 120, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 120*x^18 - 455*x^17 + 2959*x^16 - 15417*x^15 - 126061*x^14 - 215365*x^13 - 5912224*x^12 - 11246243*x^11 - 43052462*x^10 - 319740577*x^9 + 1019228952*x^8 + 10583460950*x^7 + 89634505875*x^6 + 650613452731*x^5 + 3108087294944*x^4 + 11858163795028*x^3 + 30961063038531*x^2 + 66656193013587*x + 125053657678981)
 
gp: K = bnfinit(x^20 - 4*x^19 + 120*x^18 - 455*x^17 + 2959*x^16 - 15417*x^15 - 126061*x^14 - 215365*x^13 - 5912224*x^12 - 11246243*x^11 - 43052462*x^10 - 319740577*x^9 + 1019228952*x^8 + 10583460950*x^7 + 89634505875*x^6 + 650613452731*x^5 + 3108087294944*x^4 + 11858163795028*x^3 + 30961063038531*x^2 + 66656193013587*x + 125053657678981, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 120 x^{18} - 455 x^{17} + 2959 x^{16} - 15417 x^{15} - 126061 x^{14} - 215365 x^{13} - 5912224 x^{12} - 11246243 x^{11} - 43052462 x^{10} - 319740577 x^{9} + 1019228952 x^{8} + 10583460950 x^{7} + 89634505875 x^{6} + 650613452731 x^{5} + 3108087294944 x^{4} + 11858163795028 x^{3} + 30961063038531 x^{2} + 66656193013587 x + 125053657678981 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(19867324198595974862892094631712900390625=5^{10}\cdot 4588681^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.49$
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:  $5, 4588681$
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}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{19} + \frac{35223599513682811895726069344429814137861563599137027607365360810043903127788841858374307681995417878905059694094639905177455}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{18} - \frac{35588081191322458835748807182144220464777171504515038248192653009654734195038404978218844010473429858935947909089323918585412}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{17} + \frac{567374590959481538591071194795440903869199654440047652377261331661600647892725830981234238705915376749072613987955413580376}{1526282339774067557341486082967972368600459662565886684425610483049295511220761343495453311415890564405910591480699472317439} a^{16} + \frac{24576526953282715433942561754608184836811951703290044791595491065668037511004239662925532576260818429912996876500613422624882}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{15} - \frac{15110810048161460076738770409227278364375896913430381014001270117085826441422602563012568944659300072661315033533850593316590}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{14} - \frac{1204658890027598347307756562289502279679268881940182804594252147888132418181426988536059970681080672750338137934879026076339}{3915246002029129821006420821526537815105526960495100625265696456517758050522822576792684581458154056519509778146142124640387} a^{13} - \frac{1498276361745024075912309918599944563256738342927126001287868277322838828352154546866079627190496471600741497298224423744957}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{12} + \frac{795864631280905777721001730532765291421053839025899720416913267628294520109130257900687722685142959161396566250171247199889}{4739508318245788730691983099742651039338269478494069177953211499995180798001311540327986598607239121049932889334803624564679} a^{11} + \frac{12747961652276258543819738868426992558095192137426754863656519958239244243049413596401201520584980367269749455695480063913394}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{10} - \frac{23470383378985527345974070692666605021875844016766709251158846349481762910801548785894666230157203394712595641316574532864947}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{9} - \frac{28498592416626907003134757647242385829083539956432091276063784633079663290921070422221001592854318241683010679376354028915197}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{8} - \frac{11971695316998457292811489724906862127008580885298136823682424127051167353478600661145113557502935941027000923364299873528253}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{7} - \frac{699227093817861272885926903502085825016223431134222087651427512742651162617054151262131562115838547010318033863059904650842}{1526282339774067557341486082967972368600459662565886684425610483049295511220761343495453311415890564405910591480699472317439} a^{6} - \frac{33466154218320161135566521676515774741732422918295947413438741755018870176408834552049163303004495805103162998245157584414070}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{5} + \frac{4267597577752980623351205365613293517658797007340305028623147143233772863676759187309560124792115938773654582315888529187286}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{4} + \frac{38059291971561560560949164261022799561398996602459086749371683396937656006382613100526605249173001566150926123932183615874079}{90050658046669985883147678895110369747427120091387314381111018499908435162024919266231745373537543299948724897361268866728901} a^{3} + \frac{1141985912080478932706418932506354241315020516266608763559270608017639899038027581942747597438567557913219815292171187150527}{4739508318245788730691983099742651039338269478494069177953211499995180798001311540327986598607239121049932889334803624564679} a^{2} + \frac{567488754567768829666964541184729422017606662505405974494470909084854426285208109631817477713483798829510165613613414601272}{2904859936989354383327344480487431282175068390044752076810032854835755972968545782781669205597985267740281448301976415055771} a + \frac{510064266499816860372795802570671576737905538992670150079868329908053654159812429345961659224349190670814988394159445817621}{1139881747426202352951236441710257844904140760650472333938114158226689052683859737547237283209335991138591454396978086920619}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  $11$
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:  \( 347221766867 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1022:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 189 conjugacy class representatives for t20n1022 are not computed
Character table for t20n1022 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.14339628125.1

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 32 sibling: 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.6.0.1}{6} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.1$x^{10} - 50 x^{6} + 625 x^{2} - 12500$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
4588681Data not computed