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Properties

Label 20.4.39238824049...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{28}\cdot 5^{22}\cdot 19^{10}$
Root discriminant $67.56$
Ramified primes $2, 5, 19$
Class number Not computed
Class group Not computed
Galois group $C_2\times F_5$ (as 20T13)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13944661505, 111115800, -1269028620, 2529659160, 9025, 833920, 50393340, 760, -34190, 388800, -1232184, -720, 7910, -31400, 0, -8, -295, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 295*x^16 - 8*x^15 - 31400*x^13 + 7910*x^12 - 720*x^11 - 1232184*x^10 + 388800*x^9 - 34190*x^8 + 760*x^7 + 50393340*x^6 + 833920*x^5 + 9025*x^4 + 2529659160*x^3 - 1269028620*x^2 + 111115800*x + 13944661505)
 
gp: K = bnfinit(x^20 - 295*x^16 - 8*x^15 - 31400*x^13 + 7910*x^12 - 720*x^11 - 1232184*x^10 + 388800*x^9 - 34190*x^8 + 760*x^7 + 50393340*x^6 + 833920*x^5 + 9025*x^4 + 2529659160*x^3 - 1269028620*x^2 + 111115800*x + 13944661505, 1)
 

Normalized defining polynomial

\( x^{20} - 295 x^{16} - 8 x^{15} - 31400 x^{13} + 7910 x^{12} - 720 x^{11} - 1232184 x^{10} + 388800 x^{9} - 34190 x^{8} + 760 x^{7} + 50393340 x^{6} + 833920 x^{5} + 9025 x^{4} + 2529659160 x^{3} - 1269028620 x^{2} + 111115800 x + 13944661505 \)

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:  \(3923882404992640000000000000000000000=2^{28}\cdot 5^{22}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.56$
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:  $2, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{9} + \frac{1}{4} a^{8} + \frac{1}{4} a^{6} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} + \frac{1}{4} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} + \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{12} a^{16} - \frac{1}{12} a^{15} + \frac{1}{12} a^{14} - \frac{1}{4} a^{13} - \frac{1}{4} a^{12} - \frac{1}{4} a^{11} - \frac{1}{12} a^{10} + \frac{5}{12} a^{9} - \frac{1}{4} a^{8} - \frac{1}{12} a^{7} - \frac{5}{12} a^{6} + \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{5}{12} a^{2} - \frac{1}{12} a - \frac{1}{3}$, $\frac{1}{12} a^{17} + \frac{1}{12} a^{14} - \frac{1}{4} a^{12} + \frac{1}{6} a^{11} + \frac{1}{12} a^{10} - \frac{1}{3} a^{9} + \frac{5}{12} a^{8} - \frac{1}{2} a^{7} + \frac{1}{12} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{3} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{12}$, $\frac{1}{228} a^{18} + \frac{1}{38} a^{17} + \frac{2}{57} a^{16} - \frac{25}{228} a^{15} - \frac{1}{57} a^{14} + \frac{1}{76} a^{13} - \frac{11}{114} a^{12} + \frac{1}{12} a^{11} - \frac{2}{19} a^{10} - \frac{3}{76} a^{9} + \frac{15}{38} a^{8} + \frac{35}{228} a^{7} - \frac{29}{114} a^{6} + \frac{7}{76} a^{5} - \frac{4}{57} a^{4} + \frac{67}{228} a^{3} - \frac{43}{228} a^{2} - \frac{35}{76} a + \frac{1}{57}$, $\frac{1}{79605054305162249689302073467334798218819514482786587973657874160958747776958987953805505788} a^{19} + \frac{1050721080136009360481635169160163535076049150864305453607456018246998744843118309663671}{13267509050860374948217012244555799703136585747131097995609645693493124629493164658967584298} a^{18} - \frac{426733614976560402028031708373263786551683479444448511869864723890343158943056311446788125}{19901263576290562422325518366833699554704878620696646993414468540239686944239746988451376447} a^{17} + \frac{489629011742577245979865037163045693401739551778293523391887673197238329965910672798050463}{13267509050860374948217012244555799703136585747131097995609645693493124629493164658967584298} a^{16} - \frac{115302962266964402829879469599603622633346850358856446139946605728964209703222994444147494}{2843037653755794631760788338119099936386411231528092427630638362891383849177106712635910921} a^{15} - \frac{1080803933960705375454058731798695461732238126868990577948655595254519309418925525963849028}{19901263576290562422325518366833699554704878620696646993414468540239686944239746988451376447} a^{14} + \frac{1622310633803948786771666989415024972188924759492970679512090237973135621094587216054673061}{19901263576290562422325518366833699554704878620696646993414468540239686944239746988451376447} a^{13} - \frac{338893713178897462844010215741976099054931873549086499739996784317391760776260051225065633}{39802527152581124844651036733667399109409757241393293986828937080479373888479493976902752894} a^{12} + \frac{911815883346612320444636125515160217331102978085924524471971979562684032026511745746666334}{6633754525430187474108506122277899851568292873565548997804822846746562314746582329483792149} a^{11} - \frac{810107400740685250176902766025465304583026514499478625903680621881040533677614833109934934}{19901263576290562422325518366833699554704878620696646993414468540239686944239746988451376447} a^{10} - \frac{1729545442840113872436563018236229740132297199592176126330928017414116644070534344795254963}{5686075307511589263521576676238199872772822463056184855261276725782767698354213425271821842} a^{9} - \frac{385090391807367499823098017585568928041227203835827602782221234986870496544862687349445479}{2094869850135848676034265091245652584705776696915436525622575635814703888867341788258039626} a^{8} - \frac{14064891277034037281129083575070538513279614434908681643066127798481268806378922380024337487}{39802527152581124844651036733667399109409757241393293986828937080479373888479493976902752894} a^{7} - \frac{6899504840356235722508249483745913258910950726065025404105114470077583164750604559560548646}{19901263576290562422325518366833699554704878620696646993414468540239686944239746988451376447} a^{6} - \frac{17249841804159246122151249557496085793430202867689687826178759049902915092953772074260773547}{39802527152581124844651036733667399109409757241393293986828937080479373888479493976902752894} a^{5} - \frac{8622785535007422435337995481775468891024100919124725538832741226719276760103922648038709745}{39802527152581124844651036733667399109409757241393293986828937080479373888479493976902752894} a^{4} + \frac{26682763993458657097503506814488469793400456338515385215367829488017967925051435129591979509}{79605054305162249689302073467334798218819514482786587973657874160958747776958987953805505788} a^{3} + \frac{8217174760704976300373141206814207782012944446713271118879622002417275564321849510951003575}{39802527152581124844651036733667399109409757241393293986828937080479373888479493976902752894} a^{2} + \frac{2283436605400497027566737093417969455305629056495117899311447759029945108726379877635068821}{6633754525430187474108506122277899851568292873565548997804822846746562314746582329483792149} a + \frac{5205355356534322523384227072086280938504544202650364251433070689938084706568986064799316084}{19901263576290562422325518366833699554704878620696646993414468540239686944239746988451376447}$

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 20T13):

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

\(\Q(\sqrt{95}) \), \(\Q(\sqrt{19}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{5}, \sqrt{19})\), 5.1.50000.1, 10.2.396175840000000000.1, 10.2.1980879200000000000.1, 10.2.12500000000.1

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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
2Data not computed
$5$5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
5.10.11.2$x^{10} + 5 x^{2} + 5$$10$$1$$11$$F_5$$[5/4]_{4}$
$19$19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$