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Properties

Label 20.10.5795258604...9008.1
Degree $20$
Signature $[10, 5]$
Discriminant $-\,2^{20}\cdot 11^{18}\cdot 1583^{5}$
Root discriminant $109.18$
Ramified primes $2, 11, 1583$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T130

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6758223659, -6538575428, -8525407341, -6571612740, 6434665655, 4990875340, -317626815, -341587532, -335257703, -62057688, 20230134, 4679350, 2245180, 112182, -55794, 8240, -4329, -316, 26, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^19 + 26*x^18 - 316*x^17 - 4329*x^16 + 8240*x^15 - 55794*x^14 + 112182*x^13 + 2245180*x^12 + 4679350*x^11 + 20230134*x^10 - 62057688*x^9 - 335257703*x^8 - 341587532*x^7 - 317626815*x^6 + 4990875340*x^5 + 6434665655*x^4 - 6571612740*x^3 - 8525407341*x^2 - 6538575428*x - 6758223659)
 
gp: K = bnfinit(x^20 - 2*x^19 + 26*x^18 - 316*x^17 - 4329*x^16 + 8240*x^15 - 55794*x^14 + 112182*x^13 + 2245180*x^12 + 4679350*x^11 + 20230134*x^10 - 62057688*x^9 - 335257703*x^8 - 341587532*x^7 - 317626815*x^6 + 4990875340*x^5 + 6434665655*x^4 - 6571612740*x^3 - 8525407341*x^2 - 6538575428*x - 6758223659, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{19} + 26 x^{18} - 316 x^{17} - 4329 x^{16} + 8240 x^{15} - 55794 x^{14} + 112182 x^{13} + 2245180 x^{12} + 4679350 x^{11} + 20230134 x^{10} - 62057688 x^{9} - 335257703 x^{8} - 341587532 x^{7} - 317626815 x^{6} + 4990875340 x^{5} + 6434665655 x^{4} - 6571612740 x^{3} - 8525407341 x^{2} - 6538575428 x - 6758223659 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-57952586042902321051693895129329600299008=-\,2^{20}\cdot 11^{18}\cdot 1583^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $109.18$
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:  $2, 11, 1583$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{14} - \frac{4}{11} a^{13} + \frac{3}{11} a^{12} + \frac{3}{11} a^{11} - \frac{1}{11} a^{10}$, $\frac{1}{11} a^{16} - \frac{5}{11} a^{14} - \frac{1}{11} a^{13} - \frac{5}{11} a^{12} + \frac{2}{11} a^{11} - \frac{1}{11} a^{10}$, $\frac{1}{11} a^{17} + \frac{5}{11} a^{14} - \frac{3}{11} a^{13} - \frac{5}{11} a^{12} + \frac{3}{11} a^{11} - \frac{5}{11} a^{10}$, $\frac{1}{10879} a^{18} - \frac{85}{10879} a^{17} + \frac{290}{10879} a^{16} - \frac{2}{10879} a^{15} - \frac{2476}{10879} a^{14} + \frac{3816}{10879} a^{13} - \frac{1120}{10879} a^{12} + \frac{2708}{10879} a^{11} + \frac{4355}{10879} a^{10} - \frac{239}{989} a^{9} - \frac{447}{989} a^{8} + \frac{256}{989} a^{7} - \frac{103}{989} a^{6} + \frac{13}{989} a^{5} - \frac{7}{43} a^{4} - \frac{423}{989} a^{3} - \frac{148}{989} a^{2} - \frac{80}{989} a - \frac{297}{989}$, $\frac{1}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{19} - \frac{1088823886764943422593374615526717496799322922701923161795422116319693393045011035064961259080816124}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{18} + \frac{2144443448456221978981944806070047765147908609613439477353855940839153058846476348712349112211653251014}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{17} + \frac{3638198577801545263380166448025383123940085639901635596335589544881618866403539299082887727763866089968}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{16} - \frac{2774718208313549464012322872521554062411131999827960735528427008307381378846525277928145353268467116257}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{15} + \frac{1691946465566793423067581503987169473103262125370311377278717355449396527525781272205077834279187585990}{4134078024492222305698091136481635363779827646337375935021734400847252741331437397854914175971682459991} a^{14} + \frac{8085134143764696832403419542539894447879974950782579481683961570339946426600331602789438540598156988894}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{13} + \frac{30872169661303478000497089995890555703514609103497580053834669457832630820344134176698165833100057117488}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{12} + \frac{2735885287068830742880023259782395337533623641870398817777705047864026392135324232564681115024233194501}{95083794563321113031056096139077613366936035865759646505499891219486813050623060150663026047348696579793} a^{11} - \frac{3421347576665695502599049012303006109580325004335419976798020781337882042259620515368588786585219595981}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{10} - \frac{3726060554641158106677115704287712031731865908204066919937764823672028584287415002782181085490205268029}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{9} + \frac{4716584211404822995254507959086202026946003449432769462652331191117839656628964027092889480939922337}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{8} - \frac{400901130186008578141511031435848866908575132250569098463430191283834415400696471904085794466945436245}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{7} + \frac{1179290516116526997055026034444546920040048578769086416446288775262084584363524981346818063510200804465}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{6} + \frac{770259671086295080790087583425730067001034649395497062437809168241386096391987448500244969127818157354}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{5} + \frac{3304556183666359012162280656237441318194703387982893404040898097869971361722959190121459746524099090048}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{4} + \frac{475870745457048308138347576895551443004609934890676994150284071486316687288906931395422437242248387069}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{3} - \frac{91434373049499057499395031727849173306896442978104534324196447969013413325494967518910114780680418}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a^{2} + \frac{1808792498704557560788750555522118205108926735369417475204122365576542004640431192239964788999810065486}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163} a + \frac{3047772128186053066174649450556322718300166920045158058440414290079842667749846755444918739010379184552}{8643981323938283002823281467188873942448730533250876955045444656316983004602096377333002367940790598163}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $14$
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:  \( 9182698085030 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T130:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 640
The 40 conjugacy class representatives for t20n130
Character table for t20n130 is not computed

Intermediate fields

\(\Q(\sqrt{11}) \), \(\Q(\zeta_{11})^+\), \(\Q(\zeta_{44})^+\)

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.10.0.1}{10} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }{,}\,{\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ $20$ $20$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ $20$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ $20$

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
2Data not computed
$11$11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.1$x^{10} - 11$$10$$1$$9$$C_{10}$$[\ ]_{10}$
1583Data not computed