Properties

Label 20.0.37937967949...6597.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{15}\cdot 31^{9}$
Root discriminant $10.69$
Ramified primes $3, 31$
Class number $1$
Class group Trivial
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![1, -8, 25, -26, -46, 161, -128, -127, 308, -144, -146, 193, -8, -122, 73, 25, -46, 14, 6, -5, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1)
gp: K = bnfinit(x^20 - 5*x^19 + 6*x^18 + 14*x^17 - 46*x^16 + 25*x^15 + 73*x^14 - 122*x^13 - 8*x^12 + 193*x^11 - 146*x^10 - 144*x^9 + 308*x^8 - 127*x^7 - 128*x^6 + 161*x^5 - 46*x^4 - 26*x^3 + 25*x^2 - 8*x + 1, 1)

Normalized defining polynomial

\(x^{20} \) \(\mathstrut -\mathstrut 5 x^{19} \) \(\mathstrut +\mathstrut 6 x^{18} \) \(\mathstrut +\mathstrut 14 x^{17} \) \(\mathstrut -\mathstrut 46 x^{16} \) \(\mathstrut +\mathstrut 25 x^{15} \) \(\mathstrut +\mathstrut 73 x^{14} \) \(\mathstrut -\mathstrut 122 x^{13} \) \(\mathstrut -\mathstrut 8 x^{12} \) \(\mathstrut +\mathstrut 193 x^{11} \) \(\mathstrut -\mathstrut 146 x^{10} \) \(\mathstrut -\mathstrut 144 x^{9} \) \(\mathstrut +\mathstrut 308 x^{8} \) \(\mathstrut -\mathstrut 127 x^{7} \) \(\mathstrut -\mathstrut 128 x^{6} \) \(\mathstrut +\mathstrut 161 x^{5} \) \(\mathstrut -\mathstrut 46 x^{4} \) \(\mathstrut -\mathstrut 26 x^{3} \) \(\mathstrut +\mathstrut 25 x^{2} \) \(\mathstrut -\mathstrut 8 x \) \(\mathstrut +\mathstrut 1 \)

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:  \(379379679498607236597=3^{15}\cdot 31^{9}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $10.69$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $3, 31$
magma: PrimeDivisors(Discriminant(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{6978527} a^{19} - \frac{502850}{6978527} a^{18} + \frac{2639465}{6978527} a^{17} - \frac{2706308}{6978527} a^{16} - \frac{1189948}{6978527} a^{15} - \frac{438476}{6978527} a^{14} - \frac{1096272}{6978527} a^{13} + \frac{110407}{6978527} a^{12} - \frac{3425638}{6978527} a^{11} - \frac{707323}{6978527} a^{10} - \frac{751820}{6978527} a^{9} + \frac{1184585}{6978527} a^{8} + \frac{3485122}{6978527} a^{7} - \frac{557869}{6978527} a^{6} - \frac{1191169}{6978527} a^{5} - \frac{574971}{6978527} a^{4} + \frac{918839}{6978527} a^{3} + \frac{718635}{6978527} a^{2} + \frac{68564}{6978527} a - \frac{3141208}{6978527}$

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

Class group and class number

Trivial Abelian group, order $1$

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:  \( -\frac{294846}{18413} a^{19} + \frac{1356362}{18413} a^{18} - \frac{1211607}{18413} a^{17} - \frac{4680751}{18413} a^{16} + \frac{11744765}{18413} a^{15} - \frac{2419699}{18413} a^{14} - \frac{23070963}{18413} a^{13} + \frac{26770987}{18413} a^{12} + \frac{14299518}{18413} a^{11} - \frac{52373757}{18413} a^{10} + \frac{21096924}{18413} a^{9} + \frac{53479978}{18413} a^{8} - \frac{69996438}{18413} a^{7} + \frac{6459642}{18413} a^{6} + \frac{43411115}{18413} a^{5} - \frac{29757726}{18413} a^{4} - \frac{526458}{18413} a^{3} + \frac{8417968}{18413} a^{2} - \frac{3705327}{18413} a + \frac{555268}{18413} \) (order $6$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( \frac{125833687}{6978527} a^{19} - \frac{579002682}{6978527} a^{18} + \frac{515643741}{6978527} a^{17} + \frac{2002422506}{6978527} a^{16} - \frac{5008590138}{6978527} a^{15} + \frac{1006953784}{6978527} a^{14} + \frac{9854695436}{6978527} a^{13} - \frac{11369476744}{6978527} a^{12} - \frac{6154055041}{6978527} a^{11} + \frac{22289927428}{6978527} a^{10} - \frac{8890563197}{6978527} a^{9} - \frac{22816363911}{6978527} a^{8} + \frac{29713303298}{6978527} a^{7} - \frac{2673076729}{6978527} a^{6} - \frac{18404064846}{6978527} a^{5} + \frac{12598467425}{6978527} a^{4} + \frac{154360922}{6978527} a^{3} - \frac{3559981387}{6978527} a^{2} + \frac{1601390146}{6978527} a - \frac{243752067}{6978527} \),  \( \frac{26762083}{6978527} a^{19} - \frac{117368506}{6978527} a^{18} + \frac{86709828}{6978527} a^{17} + \frac{433441327}{6978527} a^{16} - \frac{961078852}{6978527} a^{15} + \frac{43486694}{6978527} a^{14} + \frac{2010621278}{6978527} a^{13} - \frac{1961957160}{6978527} a^{12} - \frac{1555419814}{6978527} a^{11} + \frac{4200168647}{6978527} a^{10} - \frac{1092157736}{6978527} a^{9} - \frac{4696192865}{6978527} a^{8} + \frac{5151441259}{6978527} a^{7} + \frac{145118254}{6978527} a^{6} - \frac{3371710596}{6978527} a^{5} + \frac{1898278360}{6978527} a^{4} + \frac{172875141}{6978527} a^{3} - \frac{553195917}{6978527} a^{2} + \frac{232796404}{6978527} a - \frac{41793447}{6978527} \),  \( \frac{32889832}{6978527} a^{19} - \frac{108291887}{6978527} a^{18} - \frac{38389679}{6978527} a^{17} + \frac{600949327}{6978527} a^{16} - \frac{584549267}{6978527} a^{15} - \frac{1029611448}{6978527} a^{14} + \frac{2196029046}{6978527} a^{13} + \frac{144684768}{6978527} a^{12} - \frac{3717504884}{6978527} a^{11} + \frac{2590635940}{6978527} a^{10} + \frac{3396101846}{6978527} a^{9} - \frac{5806702029}{6978527} a^{8} + \frac{182070987}{6978527} a^{7} + \frac{5133858817}{6978527} a^{6} - \frac{2556534395}{6978527} a^{5} - \frac{1606071875}{6978527} a^{4} + \frac{1603082447}{6978527} a^{3} - \frac{46882479}{6978527} a^{2} - \frac{289828720}{6978527} a + \frac{85694051}{6978527} \),  \( \frac{49888150}{6978527} a^{19} - \frac{196294250}{6978527} a^{18} + \frac{67284737}{6978527} a^{17} + \frac{862418800}{6978527} a^{16} - \frac{1415541011}{6978527} a^{15} - \frac{654298751}{6978527} a^{14} + \frac{3635118793}{6978527} a^{13} - \frac{1978758124}{6978527} a^{12} - \frac{4211634325}{6978527} a^{11} + \frac{6229334093}{6978527} a^{10} + \frac{1194244276}{6978527} a^{9} - \frac{8989844543}{6978527} a^{8} + \frac{5549404629}{6978527} a^{7} + \frac{3779580930}{6978527} a^{6} - \frac{5377213314}{6978527} a^{5} + \frac{842212256}{6978527} a^{4} + \frac{1302476091}{6978527} a^{3} - \frac{609409967}{6978527} a^{2} + \frac{41000185}{6978527} a + \frac{13681465}{6978527} \),  \( \frac{21911531}{6978527} a^{19} - \frac{68310022}{6978527} a^{18} - \frac{46654449}{6978527} a^{17} + \frac{423339453}{6978527} a^{16} - \frac{312213610}{6978527} a^{15} - \frac{893649489}{6978527} a^{14} + \frac{1485656329}{6978527} a^{13} + \frac{554558403}{6978527} a^{12} - \frac{2923629334}{6978527} a^{11} + \frac{1315972066}{6978527} a^{10} + \frac{3256528943}{6978527} a^{9} - \frac{4035533311}{6978527} a^{8} - \frac{1089980004}{6978527} a^{7} + \frac{4527532959}{6978527} a^{6} - \frac{1322406858}{6978527} a^{5} - \frac{2051278318}{6978527} a^{4} + \frac{1284251518}{6978527} a^{3} + \frac{205263506}{6978527} a^{2} - \frac{302136264}{6978527} a + \frac{60791716}{6978527} \),  \( \frac{5463093}{6978527} a^{19} - \frac{47075608}{6978527} a^{18} + \frac{114802428}{6978527} a^{17} + \frac{34582096}{6978527} a^{16} - \frac{591786798}{6978527} a^{15} + \frac{767692668}{6978527} a^{14} + \frac{564049534}{6978527} a^{13} - \frac{2181637237}{6978527} a^{12} + \frac{999990953}{6978527} a^{11} + \frac{2637459188}{6978527} a^{10} - \frac{3602583653}{6978527} a^{9} - \frac{876415285}{6978527} a^{8} + \frac{5424789617}{6978527} a^{7} - \frac{3510202350}{6978527} a^{6} - \frac{1930628723}{6978527} a^{5} + \frac{3313500046}{6978527} a^{4} - \frac{865749110}{6978527} a^{3} - \frac{658884616}{6978527} a^{2} + \frac{445697455}{6978527} a - \frac{80874670}{6978527} \),  \( \frac{90156599}{6978527} a^{19} - \frac{424522713}{6978527} a^{18} + \frac{410667618}{6978527} a^{17} + \frac{1408085930}{6978527} a^{16} - \frac{3748926881}{6978527} a^{15} + \frac{1054999798}{6978527} a^{14} + \frac{7080240169}{6978527} a^{13} - \frac{8890703433}{6978527} a^{12} - \frac{3750741737}{6978527} a^{11} + \frac{16622331053}{6978527} a^{10} - \frac{7894453200}{6978527} a^{9} - \frac{16086644095}{6978527} a^{8} + \frac{23104946010}{6978527} a^{7} - \frac{3693359184}{6978527} a^{6} - \frac{13478916000}{6978527} a^{5} + \frac{10394821692}{6978527} a^{4} - \frac{517448015}{6978527} a^{3} - \frac{2746485796}{6978527} a^{2} + \frac{1383327906}{6978527} a - \frac{231596543}{6978527} \),  \( \frac{61136830}{6978527} a^{19} - \frac{261628832}{6978527} a^{18} + \frac{170092465}{6978527} a^{17} + \frac{1009733892}{6978527} a^{16} - \frac{2091572718}{6978527} a^{15} - \frac{123807846}{6978527} a^{14} + \frac{4593747192}{6978527} a^{13} - \frac{4010054276}{6978527} a^{12} - \frac{3989204187}{6978527} a^{11} + \frac{9208005573}{6978527} a^{10} - \frac{1533184215}{6978527} a^{9} - \frac{10935043913}{6978527} a^{8} + \frac{10666902952}{6978527} a^{7} + \frac{1473681256}{6978527} a^{6} - \frac{7624310740}{6978527} a^{5} + \frac{3576937525}{6978527} a^{4} + \frac{776114034}{6978527} a^{3} - \frac{1202818371}{6978527} a^{2} + \frac{405532123}{6978527} a - \frac{47186348}{6978527} \),  \( \frac{18668047}{6978527} a^{19} - \frac{68818427}{6978527} a^{18} + \frac{15181078}{6978527} a^{17} + \frac{300999886}{6978527} a^{16} - \frac{453520573}{6978527} a^{15} - \frac{236687532}{6978527} a^{14} + \frac{1143451952}{6978527} a^{13} - \frac{597123535}{6978527} a^{12} - \frac{1269009233}{6978527} a^{11} + \frac{1888293362}{6978527} a^{10} + \frac{300106238}{6978527} a^{9} - \frac{2626797553}{6978527} a^{8} + \frac{1784397631}{6978527} a^{7} + \frac{749995199}{6978527} a^{6} - \frac{1404416450}{6978527} a^{5} + \frac{589181007}{6978527} a^{4} + \frac{101068945}{6978527} a^{3} - \frac{217704765}{6978527} a^{2} + \frac{115058289}{6978527} a - \frac{19083180}{6978527} \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 181.753135766 \)
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

\(\Q(\sqrt{-3}) \), 4.0.837.1, 10.0.224415603.1

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 $20$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ $20$

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
3Data not computed
$31$31.10.9.3$x^{10} - 74431$$10$$1$$9$$C_{10}$$[\ ]_{10}$
31.10.0.1$x^{10} - x + 11$$1$$10$$0$$C_{10}$$[\ ]^{10}$