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Properties

Label 20.0.10111569723...9641.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{10}\cdot 11^{9}\cdot 19^{15}$
Root discriminant $70.83$
Ramified primes $7, 11, 19$
Class number Not computed
Class group Not computed
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![22061643433, 12852126152, 6102106147, 5939873213, 2775269509, 686312568, 260007308, 87517488, 4621179, -7923930, -1267971, 737381, 241234, -2268, -3098, 2578, 645, -2, -29, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 - 29*x^18 - 2*x^17 + 645*x^16 + 2578*x^15 - 3098*x^14 - 2268*x^13 + 241234*x^12 + 737381*x^11 - 1267971*x^10 - 7923930*x^9 + 4621179*x^8 + 87517488*x^7 + 260007308*x^6 + 686312568*x^5 + 2775269509*x^4 + 5939873213*x^3 + 6102106147*x^2 + 12852126152*x + 22061643433)
 
gp: K = bnfinit(x^20 - x^19 - 29*x^18 - 2*x^17 + 645*x^16 + 2578*x^15 - 3098*x^14 - 2268*x^13 + 241234*x^12 + 737381*x^11 - 1267971*x^10 - 7923930*x^9 + 4621179*x^8 + 87517488*x^7 + 260007308*x^6 + 686312568*x^5 + 2775269509*x^4 + 5939873213*x^3 + 6102106147*x^2 + 12852126152*x + 22061643433, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} - 29 x^{18} - 2 x^{17} + 645 x^{16} + 2578 x^{15} - 3098 x^{14} - 2268 x^{13} + 241234 x^{12} + 737381 x^{11} - 1267971 x^{10} - 7923930 x^{9} + 4621179 x^{8} + 87517488 x^{7} + 260007308 x^{6} + 686312568 x^{5} + 2775269509 x^{4} + 5939873213 x^{3} + 6102106147 x^{2} + 12852126152 x + 22061643433 \)

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:  \(10111569723784930165426015116328899641=7^{10}\cdot 11^{9}\cdot 19^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $70.83$
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:  $7, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{7} a^{16} + \frac{2}{7} a^{15} + \frac{2}{7} a^{12} - \frac{3}{7} a^{11} - \frac{2}{7} a^{10} - \frac{2}{7} a^{9} - \frac{1}{7} a^{8} - \frac{3}{7} a^{7} - \frac{1}{7} a^{6} - \frac{3}{7} a^{5} - \frac{3}{7} a^{3} - \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{7} a^{17} + \frac{3}{7} a^{15} + \frac{2}{7} a^{13} - \frac{3}{7} a^{11} + \frac{2}{7} a^{10} + \frac{3}{7} a^{9} - \frac{1}{7} a^{8} - \frac{2}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{1}{7} a^{3} - \frac{3}{7} a^{2} + \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{119} a^{18} - \frac{1}{119} a^{17} - \frac{4}{119} a^{16} - \frac{10}{119} a^{15} - \frac{33}{119} a^{14} + \frac{33}{119} a^{13} + \frac{4}{119} a^{12} + \frac{19}{119} a^{11} + \frac{36}{119} a^{10} - \frac{46}{119} a^{9} + \frac{41}{119} a^{8} - \frac{20}{119} a^{7} - \frac{4}{17} a^{6} + \frac{19}{119} a^{5} + \frac{23}{119} a^{4} - \frac{9}{119} a^{3} - \frac{10}{119} a^{2} + \frac{58}{119} a + \frac{46}{119}$, $\frac{1}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{19} - \frac{37519960713237090616124962930497179755637010971343265649010970970584221240020513834775619791}{91362997670018087352829412758637108441910332763190131900502208212732467994746247065974822866903} a^{18} - \frac{14908691147746152423661520868020955713367722186773591012797571977031401053370160110680503331256}{3432638341030679567684876507931651360031773930959857812833154394278377011802608996907339773427927} a^{17} - \frac{708238765158910192944144024636082900233347172014600767524115650884744155753236920109048417135440}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{16} - \frac{9716597634630961517459933817494933259498008115534903937041079101105923251119911187887183105509009}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{15} - \frac{33018425982116828063124197431300585771735357950546511162474421315641286679257620719904221447628}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{14} - \frac{2882119589689803583999696655520087827782120542676742615223879399692381544415699967487383176703901}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{13} - \frac{341432886819809965458084327396508912329551450547113328127907025837803663012273769498107994794040}{1413439316894985704340831503265974089424848089218764981754828279996978769565780175197139906705617} a^{12} - \frac{2985790802694584755163360118907953860564515975255028467943760513242837949193141181171173330161314}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{11} + \frac{7337527313398128375359415063321480362192453679405021841148079693895601067082891761942208475129217}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{10} - \frac{1697546052785028743297482767615714809449589791005522218008846209132898990749877077280221053940448}{3432638341030679567684876507931651360031773930959857812833154394278377011802608996907339773427927} a^{9} - \frac{6616637257909397170595494115816981606495552018306904259932516245976561075851147737925909609844074}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{8} + \frac{11777424200440090532208411245925115581088766354212391815072019313063722515593556180744591043827478}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{7} - \frac{6347558700485342254507138543696277894958956889848792191956353351788371438942186860822730707942933}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{6} - \frac{5373573752763421354864352624776938364328173924147189591604028130893268242952587277350405373322696}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{5} - \frac{664761524306387673142335713744432063522403249242111346518445881183069841412421485725631942179283}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489} a^{4} + \frac{175624815385868910641129489524433904866042095042067278737840098894204689167211697727097742134726}{3432638341030679567684876507931651360031773930959857812833154394278377011802608996907339773427927} a^{3} - \frac{236311445036311394712705631476265366607539401540950929758239046127350634156201976403665667727148}{3432638341030679567684876507931651360031773930959857812833154394278377011802608996907339773427927} a^{2} + \frac{61955974720326816017726667302794757710109738112233722308409668533155271435300994132944317036001}{1413439316894985704340831503265974089424848089218764981754828279996978769565780175197139906705617} a - \frac{6033441905098433393764063013986258194111618975317894985481252558482183680318697034740118496249959}{24028468387214756973794135555521559520222417516719004689832080759948639082618262978351378413995489}$

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:  $9$
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:  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

$C_5\times C_5:D_4$ (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

\(\Q(\sqrt{-19}) \), 4.0.3697001.1, 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 ${\href{/LocalNumberField/2.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/5.5.0.1}{5} }^{4}$ R 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 ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ $20$ $20$

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
$7$7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.10.5.2$x^{10} - 2401 x^{2} + 67228$$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