↑

Properties

Label 20.0.34957327565...0625.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{10}\cdot 11^{9}\cdot 19^{15}$
Root discriminant $59.87$
Ramified primes $5, 11, 19$
Class number $44$ (GRH)
Class group $[2, 22]$ (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![42152299, 330515948, 696443374, 94712168, 60299602, 43648056, 31657670, 14990118, 3626130, 1419624, 741564, 165614, 53419, 4515, 4597, 925, 132, -2, 28, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 28*x^18 - 2*x^17 + 132*x^16 + 925*x^15 + 4597*x^14 + 4515*x^13 + 53419*x^12 + 165614*x^11 + 741564*x^10 + 1419624*x^9 + 3626130*x^8 + 14990118*x^7 + 31657670*x^6 + 43648056*x^5 + 60299602*x^4 + 94712168*x^3 + 696443374*x^2 + 330515948*x + 42152299)
 
gp: K = bnfinit(x^20 - x^19 + 28*x^18 - 2*x^17 + 132*x^16 + 925*x^15 + 4597*x^14 + 4515*x^13 + 53419*x^12 + 165614*x^11 + 741564*x^10 + 1419624*x^9 + 3626130*x^8 + 14990118*x^7 + 31657670*x^6 + 43648056*x^5 + 60299602*x^4 + 94712168*x^3 + 696443374*x^2 + 330515948*x + 42152299, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 28 x^{18} - 2 x^{17} + 132 x^{16} + 925 x^{15} + 4597 x^{14} + 4515 x^{13} + 53419 x^{12} + 165614 x^{11} + 741564 x^{10} + 1419624 x^{9} + 3626130 x^{8} + 14990118 x^{7} + 31657670 x^{6} + 43648056 x^{5} + 60299602 x^{4} + 94712168 x^{3} + 696443374 x^{2} + 330515948 x + 42152299 \)

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:  \(349573275653036803400564234462890625=5^{10}\cdot 11^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $59.87$
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:  $5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} - \frac{3}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{2}{7} a^{7} + \frac{3}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{4} - \frac{2}{7} a^{3} - \frac{1}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{7} a^{16} - \frac{1}{7} a^{14} + \frac{2}{7} a^{13} - \frac{1}{7} a^{10} + \frac{1}{7} a^{9} - \frac{2}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} + \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{1}{7} a^{2} - \frac{2}{7} a$, $\frac{1}{7} a^{17} - \frac{1}{7} a^{14} + \frac{1}{7} a^{13} - \frac{1}{7} a^{12} + \frac{2}{7} a^{11} - \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} - \frac{2}{7} a^{6} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} - \frac{3}{7} a$, $\frac{1}{119} a^{18} + \frac{5}{119} a^{17} - \frac{3}{119} a^{15} - \frac{5}{119} a^{14} - \frac{19}{119} a^{13} + \frac{13}{119} a^{12} + \frac{1}{7} a^{11} + \frac{16}{119} a^{10} - \frac{40}{119} a^{9} - \frac{39}{119} a^{8} - \frac{2}{7} a^{7} - \frac{37}{119} a^{6} + \frac{10}{119} a^{5} + \frac{11}{119} a^{4} + \frac{59}{119} a^{3} + \frac{12}{119} a^{2} + \frac{33}{119} a$, $\frac{1}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{19} - \frac{339454672887835031859553367711791383929118474026374548641628284393241949167248903}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{18} + \frac{4966713749920538796456274309721508379549400748802743737026204061561115174126374932}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{17} + \frac{9055722894660398242007794292942371722352872884438176310332115099317002519951199279}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{16} + \frac{3780154829273123946135670965028259214731301021307332654148117711462641049398409730}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{15} - \frac{47872628207816528335631494811357359445975283596868823872319712750185748583075099285}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{14} - \frac{34790715904979082464770134953733843227971115708048189931325232735580367060899569363}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{13} + \frac{84384166180082854579064592241120836916366678147916618897708052136731203487047454280}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{12} - \frac{663749237571928774826451491153480325462185842636339541621814296718235264594573931}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{11} - \frac{14442329894732662443149660236786824411424634352500977003470213312732175936674018475}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{10} - \frac{11537265446793043244183765232546932197636311924218041906157172705997017861754934347}{25408789725209269342872392912915058766896613193186077102606907524700499869976493199} a^{9} + \frac{4175928433737196607440824125544184126878649576026822752172982448394874455149216751}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{8} + \frac{69061291729379344539313999329837789981791375611997401733057022159304174851206514479}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{7} + \frac{52003928947501660058515226163816996797650901846856241476820933097473838641816672413}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{6} + \frac{30242023548908691622961711605835474898759121747938100477110825081297346597341779609}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{5} + \frac{6950324422200538035035412070813634599244077094376092163815598716458943695176205351}{25408789725209269342872392912915058766896613193186077102606907524700499869976493199} a^{4} - \frac{53739339195765136496089033318033719898131685442471342640344224833097498604445044923}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{3} + \frac{35867479196282244908578497601834431422200175099362352753335638425025750037323387800}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a^{2} - \frac{3991131677466146157768771507191909445332239754932866608456387146197938722162729327}{177861528076464885400106750390405411368276292352302539718248352672903499089835452393} a + \frac{359984439301674626856853860663144996904657063607057071182069187398870604632390492}{1494634689718192314286611347818532868640977246658004535447465148511794109998617247}$

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

Class group and class number

$C_{2}\times C_{22}$, which has order $44$ (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:  \( 355564701.5857714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-19}) \), 4.0.1886225.3, 10.0.36252565459.1

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 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{10}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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$5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
5.10.5.2$x^{10} - 625 x^{2} + 6250$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
$11$11.10.0.1$x^{10} + x^{2} - x + 6$$1$$10$$0$$C_{10}$$[\ ]^{10}$
11.10.9.4$x^{10} - 99$$10$$1$$9$$C_{10}$$[\ ]_{10}$
19Data not computed