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Properties

Label 20.4.10063630674...3125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{31}\cdot 43^{10}$
Root discriminant $79.46$
Ramified primes $5, 43$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1412257296521, -158789506050, 175486910000, 18580850960, -112341446200, 372998712, 14514361470, -1679254355, -425930625, 234681130, -54740221, -1764255, 2435570, -378895, 40505, -5077, -1045, 555, -40, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 - 40*x^18 + 555*x^17 - 1045*x^16 - 5077*x^15 + 40505*x^14 - 378895*x^13 + 2435570*x^12 - 1764255*x^11 - 54740221*x^10 + 234681130*x^9 - 425930625*x^8 - 1679254355*x^7 + 14514361470*x^6 + 372998712*x^5 - 112341446200*x^4 + 18580850960*x^3 + 175486910000*x^2 - 158789506050*x + 1412257296521)
 
gp: K = bnfinit(x^20 - 5*x^19 - 40*x^18 + 555*x^17 - 1045*x^16 - 5077*x^15 + 40505*x^14 - 378895*x^13 + 2435570*x^12 - 1764255*x^11 - 54740221*x^10 + 234681130*x^9 - 425930625*x^8 - 1679254355*x^7 + 14514361470*x^6 + 372998712*x^5 - 112341446200*x^4 + 18580850960*x^3 + 175486910000*x^2 - 158789506050*x + 1412257296521, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} - 40 x^{18} + 555 x^{17} - 1045 x^{16} - 5077 x^{15} + 40505 x^{14} - 378895 x^{13} + 2435570 x^{12} - 1764255 x^{11} - 54740221 x^{10} + 234681130 x^{9} - 425930625 x^{8} - 1679254355 x^{7} + 14514361470 x^{6} + 372998712 x^{5} - 112341446200 x^{4} + 18580850960 x^{3} + 175486910000 x^{2} - 158789506050 x + 1412257296521 \)

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:  \(100636306746323821134865283966064453125=5^{31}\cdot 43^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $79.46$
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:  $5, 43$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{18} - \frac{1}{12} a^{17} - \frac{1}{12} a^{16} - \frac{1}{6} a^{15} + \frac{1}{6} a^{14} - \frac{1}{12} a^{13} - \frac{1}{4} a^{12} + \frac{1}{3} a^{11} - \frac{1}{4} a^{10} - \frac{5}{12} a^{9} - \frac{1}{12} a^{7} + \frac{5}{12} a^{6} - \frac{1}{3} a^{5} + \frac{1}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} + \frac{1}{6} a - \frac{5}{12}$, $\frac{1}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{19} + \frac{3193320932620953431677156420171432558141340435621542973408139831014565722560709896021956879660862483431091013297}{136524673300481896051277828575491615179842405311690262143260321337726287803751527852189990492790857499977802140497} a^{18} + \frac{16810476808750061835162686512246373266522647337447985625812350442670376852814982117443722925101763992070319559421}{91016448866987930700851885716994410119894936874460174762173547558484191869167685234793326995193904999985201426998} a^{17} + \frac{72724325331724675160557329633034515544834866453883357834288227609208636714383011334255788130959227755736286212823}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{16} + \frac{87174053280473503352205226057757969375925339897203009716658244004714601014397278615157729248558929967156246771}{9255910054269959054323920581389262046091010529606119467339682802557714427372984939131524779172261525422223873932} a^{15} + \frac{67481138839003953662060935418705920820678344232554119659671838037448148611014762910381242988862574607288266787339}{364065795467951722803407542867977640479579747497840699048694190233936767476670740939173307980775619999940805707992} a^{14} - \frac{321704067739830922802386066624711932943485922240717697874276911017751347415028056374692742785373718962102258577}{9255910054269959054323920581389262046091010529606119467339682802557714427372984939131524779172261525422223873932} a^{13} + \frac{133548938066349703176732377086350586578498133892613650591434835690875219377145460004391625926985693780599056468271}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{12} - \frac{129621282503408391809797968908357051025922886733349781885121266765403561133110915862674158122406376552440128868203}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{11} + \frac{136305219676319606049738835293713111342033102062667491911636454257589885301958463607807399783279435150582644527263}{546098693201927584205111314301966460719369621246761048573041285350905151215006111408759961971163429999911208561988} a^{10} + \frac{270062724453590841269152661702093962444427805042259893895354959521493294955911493374833132785931291987359104609461}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{9} - \frac{240318855226271201542926344238381236935464982379025607890910935872487581176761591400147636150721546877048875315341}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{8} + \frac{43217501903916769604739139073345119763371418916418122539385061284879746616996377553299001848113943173560318057275}{182032897733975861401703771433988820239789873748920349524347095116968383738335370469586653990387809999970402853996} a^{7} + \frac{29549141421462304007268123689569093907308163436263065675187274976560697326551248664192923250518165255749504797289}{364065795467951722803407542867977640479579747497840699048694190233936767476670740939173307980775619999940805707992} a^{6} - \frac{485710857333334885446433433231046148351311146010444698128417183085532100754250907396666209752319872076571020198139}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{5} + \frac{369425806517040690306524859580564236401525848082481503213022041950186142847358131841347090501388095795940289989901}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{4} + \frac{178148830437358433500508094678902417608389377328704700698484730667937725902435686203584567777654561139386502915399}{364065795467951722803407542867977640479579747497840699048694190233936767476670740939173307980775619999940805707992} a^{3} - \frac{145089176311619215515794270623209331683265550382253828585211103283994133213079711210668416833840493721485245804459}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a^{2} - \frac{376877797358833439423289821127626686293842458067576281306912792548162630141681624077446757443925674797382273891531}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976} a - \frac{239067869267453050948478636260660303587846653923054223805048068618204899654593922604350320981273647997794461875169}{1092197386403855168410222628603932921438739242493522097146082570701810302430012222817519923942326859999822417123976}$

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:  $11$
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:  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

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.4.231125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$43$43.4.2.2$x^{4} - 43 x^{2} + 5547$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
43.8.4.1$x^{8} + 73960 x^{4} - 79507 x^{2} + 1367520400$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$