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Properties

Label 20.0.44181154111...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{16}\cdot 6029^{7}$
Root discriminant $76.25$
Ramified primes $5, 6029$
Class number $110764$ (GRH)
Class group $[2, 55382]$ (GRH)
Galois group 20T375

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5938372789, -466191748, 2425731912, -534644577, 334834464, -275793293, 23769237, -61891709, 7706052, -7038910, 2904154, -470435, 520328, -24462, 50567, -1336, 2786, -56, 82, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 82*x^18 - 56*x^17 + 2786*x^16 - 1336*x^15 + 50567*x^14 - 24462*x^13 + 520328*x^12 - 470435*x^11 + 2904154*x^10 - 7038910*x^9 + 7706052*x^8 - 61891709*x^7 + 23769237*x^6 - 275793293*x^5 + 334834464*x^4 - 534644577*x^3 + 2425731912*x^2 - 466191748*x + 5938372789)
 
gp: K = bnfinit(x^20 - x^19 + 82*x^18 - 56*x^17 + 2786*x^16 - 1336*x^15 + 50567*x^14 - 24462*x^13 + 520328*x^12 - 470435*x^11 + 2904154*x^10 - 7038910*x^9 + 7706052*x^8 - 61891709*x^7 + 23769237*x^6 - 275793293*x^5 + 334834464*x^4 - 534644577*x^3 + 2425731912*x^2 - 466191748*x + 5938372789, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 82 x^{18} - 56 x^{17} + 2786 x^{16} - 1336 x^{15} + 50567 x^{14} - 24462 x^{13} + 520328 x^{12} - 470435 x^{11} + 2904154 x^{10} - 7038910 x^{9} + 7706052 x^{8} - 61891709 x^{7} + 23769237 x^{6} - 275793293 x^{5} + 334834464 x^{4} - 534644577 x^{3} + 2425731912 x^{2} - 466191748 x + 5938372789 \)

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:  \(44181154111269498973595933380126953125=5^{16}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.25$
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:  $5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{19} + \frac{778007780187889210590854187525828216446507919900035137993948772386897091411497598}{2302059552298669980369178130950090114824850441055624213506671086372629452403853737} a^{18} + \frac{13327800119398781414890940118455896547069339871014094293548296072378833051677737853}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{17} + \frac{756848661947023476411580955324612229780109614669720085095169677520532299319739766}{2302059552298669980369178130950090114824850441055624213506671086372629452403853737} a^{16} - \frac{10530566646527357613679525018708073591942831933617508790576455027153397264280616048}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{15} + \frac{181023025540188335754478145045271020284247832351068017138328683532925500849857950}{695971492555411854530216644240724918435419900784258483153179630763818206540699967} a^{14} - \frac{2027434687509144117335229691658088035038262513542611719119418869749259160729304713}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{13} - \frac{1746742118697512107121941283581898957107771031147593755914598525352756516904560109}{4275253454268958534971330814621595927531865104817587825083817731834883268750014083} a^{12} - \frac{7065375612119042399225923551048357864011260214725470423054477426053642751003216319}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{11} + \frac{14931644365023789510796342348566752080428026235303442687279641703906805166391486841}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{10} + \frac{320091423423849703663326245527662656469138397009789415347982510813998859622329081}{2302059552298669980369178130950090114824850441055624213506671086372629452403853737} a^{9} + \frac{12190024000268594261684257772817890343331677605960326129696223139120870268702117061}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{8} - \frac{5870400577525126699206772031076289826921628961721033595314411410979439568656370980}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{7} + \frac{9475561572617360235351670884503721785095553105291304337533760071706866325898867086}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{6} - \frac{221180772037539292597074777283552963136245103424966517058108883015621612408346626}{695971492555411854530216644240724918435419900784258483153179630763818206540699967} a^{5} - \frac{1395245608112300482977020589187555953946018401312165731144889081646555773108820962}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{4} - \frac{13422030875141066508588171208819857113506989062154042544551229517040128488383262761}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{3} - \frac{1898393633103916798940660013866990550801384346791454477135351024265749848223454308}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a^{2} - \frac{5605139475902250156176064230556049778051774530904994101343774116001520638940473364}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581} a + \frac{11925414344174718293788263502414698357283460812788007213838116311207029389948317491}{29926774179882709744799315702351171492723055733723114775586724122844182881250098581}$

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

Class group and class number

$C_{2}\times C_{55382}$, which has order $110764$ (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:  \( -1 \) (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:  \( 341439.528105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T375:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7680
The 48 conjugacy class representatives for t20n375
Character table for t20n375 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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

Sibling fields

Degree 20 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 $20$ $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
6029Data not computed