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Properties

Label 20.4.65971085909...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{22}\cdot 3^{16}\cdot 5^{37}\cdot 11^{5}\cdot 89^{10}$
Root discriminant $1741.68$
Ramified primes $2, 3, 5, 11, 89$
Class number Not computed
Class group Not computed
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-64435301225, 27753840000, 18457557000, 18578652000, 3385257125, -4560186240, 20076000, 289192800, 157270150, -86400, -25260624, -357600, 2639530, -28800, -164640, -480, 6035, 0, -120, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 120*x^18 + 6035*x^16 - 480*x^15 - 164640*x^14 - 28800*x^13 + 2639530*x^12 - 357600*x^11 - 25260624*x^10 - 86400*x^9 + 157270150*x^8 + 289192800*x^7 + 20076000*x^6 - 4560186240*x^5 + 3385257125*x^4 + 18578652000*x^3 + 18457557000*x^2 + 27753840000*x - 64435301225)
 
gp: K = bnfinit(x^20 - 120*x^18 + 6035*x^16 - 480*x^15 - 164640*x^14 - 28800*x^13 + 2639530*x^12 - 357600*x^11 - 25260624*x^10 - 86400*x^9 + 157270150*x^8 + 289192800*x^7 + 20076000*x^6 - 4560186240*x^5 + 3385257125*x^4 + 18578652000*x^3 + 18457557000*x^2 + 27753840000*x - 64435301225, 1)
 

Normalized defining polynomial

\( x^{20} - 120 x^{18} + 6035 x^{16} - 480 x^{15} - 164640 x^{14} - 28800 x^{13} + 2639530 x^{12} - 357600 x^{11} - 25260624 x^{10} - 86400 x^{9} + 157270150 x^{8} + 289192800 x^{7} + 20076000 x^{6} - 4560186240 x^{5} + 3385257125 x^{4} + 18578652000 x^{3} + 18457557000 x^{2} + 27753840000 x - 64435301225 \)

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:  \(65971085909696290318640396673392028808593750000000000000000000000=2^{22}\cdot 3^{16}\cdot 5^{37}\cdot 11^{5}\cdot 89^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1741.68$
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:  $2, 3, 5, 11, 89$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4}$, $\frac{1}{4} a^{5} - \frac{1}{4} a$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{8} a^{2} + \frac{1}{8}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{8} a$, $\frac{1}{16} a^{8} - \frac{1}{8} a^{4} + \frac{1}{16}$, $\frac{1}{16} a^{9} - \frac{1}{8} a^{5} + \frac{1}{16} a$, $\frac{1}{32} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{6} + \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{1}{32}$, $\frac{1}{64} a^{11} - \frac{1}{64} a^{10} + \frac{1}{64} a^{9} - \frac{1}{64} a^{8} - \frac{1}{32} a^{7} + \frac{1}{32} a^{6} - \frac{1}{32} a^{5} + \frac{1}{32} a^{4} + \frac{1}{64} a^{3} - \frac{1}{64} a^{2} + \frac{1}{64} a - \frac{1}{64}$, $\frac{1}{64} a^{12} + \frac{1}{64} a^{8} - \frac{5}{64} a^{4} + \frac{3}{64}$, $\frac{1}{128} a^{13} - \frac{1}{128} a^{12} - \frac{3}{128} a^{9} + \frac{3}{128} a^{8} + \frac{3}{128} a^{5} - \frac{3}{128} a^{4} - \frac{1}{128} a + \frac{1}{128}$, $\frac{1}{128} a^{14} - \frac{1}{128} a^{12} + \frac{1}{128} a^{10} - \frac{1}{128} a^{8} - \frac{5}{128} a^{6} + \frac{5}{128} a^{4} + \frac{3}{128} a^{2} - \frac{3}{128}$, $\frac{1}{6400} a^{15} - \frac{1}{256} a^{14} + \frac{1}{1280} a^{13} - \frac{1}{256} a^{12} - \frac{3}{1280} a^{11} + \frac{3}{1280} a^{10} + \frac{17}{1280} a^{9} + \frac{7}{256} a^{8} + \frac{31}{1280} a^{7} - \frac{11}{256} a^{6} - \frac{249}{6400} a^{5} - \frac{11}{256} a^{4} - \frac{21}{256} a^{3} + \frac{37}{256} a^{2} - \frac{9}{256} a - \frac{359}{1280}$, $\frac{1}{12800} a^{16} + \frac{3}{1280} a^{14} + \frac{1}{1280} a^{12} + \frac{1}{320} a^{11} + \frac{11}{1280} a^{10} + \frac{1}{64} a^{9} - \frac{11}{640} a^{8} + \frac{1}{32} a^{7} + \frac{213}{6400} a^{6} + \frac{1}{32} a^{5} + \frac{19}{256} a^{4} + \frac{1}{64} a^{3} - \frac{19}{256} a^{2} - \frac{63}{320} a - \frac{45}{512}$, $\frac{1}{1011200} a^{17} - \frac{3}{1011200} a^{16} + \frac{11}{505600} a^{15} - \frac{349}{101120} a^{14} - \frac{183}{101120} a^{13} + \frac{201}{101120} a^{12} + \frac{591}{101120} a^{11} - \frac{209}{20224} a^{10} + \frac{121}{10112} a^{9} + \frac{1423}{50560} a^{8} + \frac{25193}{505600} a^{7} - \frac{8539}{505600} a^{6} + \frac{5971}{505600} a^{5} + \frac{2411}{20224} a^{4} - \frac{2519}{20224} a^{3} - \frac{5807}{101120} a^{2} - \frac{83393}{202240} a - \frac{97173}{202240}$, $\frac{1}{259078582082547753500185600} a^{18} - \frac{8237158761990384269}{129539291041273876750092800} a^{17} + \frac{991454414252599451857}{51815716416509550700037120} a^{16} - \frac{3606611365166170768189}{64769645520636938375046400} a^{15} - \frac{16423573579220817648487}{12953929104127387675009280} a^{14} - \frac{16140132643265192551951}{12953929104127387675009280} a^{13} + \frac{78023689330105377365997}{12953929104127387675009280} a^{12} + \frac{79428171348034071151943}{12953929104127387675009280} a^{11} - \frac{333691954118479333389221}{25907858208254775350018560} a^{10} - \frac{15191648870186189086593}{647696455206369383750464} a^{9} - \frac{1682537019657481773977437}{129539291041273876750092800} a^{8} - \frac{4045149965217042500613967}{64769645520636938375046400} a^{7} - \frac{169099813078628209598451}{12953929104127387675009280} a^{6} + \frac{1093257671213992250161571}{64769645520636938375046400} a^{5} + \frac{270423039550745854010573}{2590785820825477535001856} a^{4} + \frac{2565629413308537387275669}{12953929104127387675009280} a^{3} - \frac{6699117193652395875387163}{51815716416509550700037120} a^{2} + \frac{5013688163180592584801}{19262348110226598773248} a + \frac{17692997680555780966236009}{51815716416509550700037120}$, $\frac{1}{18143424205294018364374208633029121034112000} a^{19} + \frac{366538055931093}{1814342420529401836437420863302912103411200} a^{18} + \frac{227440135985797840781992994647398129}{725736968211760734574968345321164841364480} a^{17} + \frac{1398892128801057262524067993371564193}{45358560513235045910935521582572802585280} a^{16} - \frac{54774407024760013928499165000512155193}{907171210264700918218710431651456051705600} a^{15} - \frac{2996940405574010422658760691324694696303}{907171210264700918218710431651456051705600} a^{14} - \frac{2739744127050661300728721569506111583317}{907171210264700918218710431651456051705600} a^{13} - \frac{471674090016952147210355066220892652821}{181434242052940183643742086330291210341120} a^{12} + \frac{3874374123766849139336675608321726174903}{1814342420529401836437420863302912103411200} a^{11} + \frac{1000927574240771781729613006614622044381}{90717121026470091821871043165145605170560} a^{10} - \frac{116411778249368625718613703880122433165837}{9071712102647009182187104316514560517056000} a^{9} + \frac{27778651889691993654683360246797207364399}{907171210264700918218710431651456051705600} a^{8} - \frac{42042984043085422753271787379039585673049}{907171210264700918218710431651456051705600} a^{7} - \frac{9743670552500482894450149368576249200493}{181434242052940183643742086330291210341120} a^{6} - \frac{31825826135525676216095378062444110270381}{907171210264700918218710431651456051705600} a^{5} + \frac{34265193269463335890058172346625935148269}{907171210264700918218710431651456051705600} a^{4} + \frac{127606044224467639719960894295332660846441}{725736968211760734574968345321164841364480} a^{3} - \frac{26529055952906359323842924689915313580773}{362868484105880367287484172660582420682240} a^{2} + \frac{34735531388610101002039808359171461014765}{145147393642352146914993669064232968272896} a + \frac{60303126196722445514768888535197391299429}{181434242052940183643742086330291210341120}$

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

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{89}) \), 4.4.1742620.2, 5.1.2531250000.22, 10.2.35778334028211914062500000000.1

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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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
$2$2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.5.4.1$x^{5} - 2$$5$$1$$4$$F_5$$[\ ]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.5.9.1$x^{5} + 5$$5$$1$$9$$F_5$$[9/4]_{4}$
5.5.9.1$x^{5} + 5$$5$$1$$9$$F_5$$[9/4]_{4}$
5.10.19.1$x^{10} + 5$$10$$1$$19$$F_5$$[9/4]_{4}$
$11$11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
$89$89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
89.4.2.1$x^{4} + 979 x^{2} + 285156$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$