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Properties

Label 20.4.16978187713...0000.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{33}\cdot 3^{17}\cdot 5^{38}\cdot 29^{10}$
Root discriminant $915.15$
Ramified primes $2, 3, 5, 29$
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![-189670693824, 26672716800, 171320650560, -11607321600, 55152558720, -10520047440, 4758606720, 1323298800, 393756960, -79322400, -35378423, 3272400, 3014885, -82800, -165830, -1440, 5770, 0, -115, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 115*x^18 + 5770*x^16 - 1440*x^15 - 165830*x^14 - 82800*x^13 + 3014885*x^12 + 3272400*x^11 - 35378423*x^10 - 79322400*x^9 + 393756960*x^8 + 1323298800*x^7 + 4758606720*x^6 - 10520047440*x^5 + 55152558720*x^4 - 11607321600*x^3 + 171320650560*x^2 + 26672716800*x - 189670693824)
 
gp: K = bnfinit(x^20 - 115*x^18 + 5770*x^16 - 1440*x^15 - 165830*x^14 - 82800*x^13 + 3014885*x^12 + 3272400*x^11 - 35378423*x^10 - 79322400*x^9 + 393756960*x^8 + 1323298800*x^7 + 4758606720*x^6 - 10520047440*x^5 + 55152558720*x^4 - 11607321600*x^3 + 171320650560*x^2 + 26672716800*x - 189670693824, 1)
 

Normalized defining polynomial

\( x^{20} - 115 x^{18} + 5770 x^{16} - 1440 x^{15} - 165830 x^{14} - 82800 x^{13} + 3014885 x^{12} + 3272400 x^{11} - 35378423 x^{10} - 79322400 x^{9} + 393756960 x^{8} + 1323298800 x^{7} + 4758606720 x^{6} - 10520047440 x^{5} + 55152558720 x^{4} - 11607321600 x^{3} + 171320650560 x^{2} + 26672716800 x - 189670693824 \)

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:  \(169781877136459328352384375000000000000000000000000000000000=2^{33}\cdot 3^{17}\cdot 5^{38}\cdot 29^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $915.15$
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:  $2, 3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4}$, $\frac{1}{12} a^{9} - \frac{1}{4} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{300} a^{10} + \frac{1}{10} a^{8} - \frac{1}{20} a^{6} + \frac{1}{10} a^{5} + \frac{1}{30} a^{4} - \frac{1}{2} a^{3} + \frac{1}{10} a^{2} - \frac{2}{25}$, $\frac{1}{600} a^{11} + \frac{1}{120} a^{9} + \frac{9}{40} a^{7} - \frac{1}{5} a^{6} - \frac{13}{120} a^{5} - \frac{11}{30} a^{3} - \frac{1}{25} a$, $\frac{1}{1200} a^{12} - \frac{1}{1200} a^{11} - \frac{1}{1200} a^{10} + \frac{3}{80} a^{9} - \frac{3}{80} a^{8} + \frac{3}{80} a^{7} - \frac{31}{240} a^{6} - \frac{53}{240} a^{5} - \frac{7}{30} a^{4} - \frac{2}{5} a^{3} + \frac{2}{25} a^{2} + \frac{1}{50} a - \frac{19}{50}$, $\frac{1}{1200} a^{13} + \frac{1}{120} a^{9} - \frac{1}{10} a^{8} + \frac{2}{15} a^{7} - \frac{13}{80} a^{5} + \frac{47}{150} a^{3} - \frac{2}{5} a - \frac{1}{2}$, $\frac{1}{1200} a^{14} - \frac{1}{600} a^{10} - \frac{1}{60} a^{9} + \frac{1}{12} a^{8} - \frac{1}{80} a^{6} - \frac{1}{20} a^{5} - \frac{11}{300} a^{4} - \frac{1}{6} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2} a + \frac{6}{25}$, $\frac{1}{7200} a^{15} - \frac{1}{2400} a^{14} + \frac{1}{3600} a^{13} - \frac{1}{3600} a^{11} - \frac{1}{1200} a^{10} + \frac{7}{180} a^{9} + \frac{1}{12} a^{8} - \frac{59}{1440} a^{7} - \frac{3}{160} a^{6} - \frac{757}{3600} a^{5} + \frac{19}{150} a^{4} + \frac{23}{50} a^{3} + \frac{1}{20} a^{2} - \frac{1}{100} a + \frac{1}{50}$, $\frac{1}{93600} a^{16} - \frac{1}{31200} a^{15} + \frac{1}{46800} a^{14} + \frac{1}{5200} a^{13} + \frac{7}{23400} a^{12} - \frac{1}{1560} a^{11} + \frac{29}{46800} a^{10} - \frac{113}{3120} a^{9} - \frac{113}{18720} a^{8} + \frac{19}{2080} a^{7} - \frac{101}{23400} a^{6} - \frac{1379}{7800} a^{5} + \frac{233}{3900} a^{4} + \frac{109}{300} a^{3} + \frac{43}{100} a^{2} + \frac{5}{13} a - \frac{94}{325}$, $\frac{1}{187200} a^{17} - \frac{7}{187200} a^{15} + \frac{1}{7800} a^{14} - \frac{37}{93600} a^{13} + \frac{1}{7800} a^{12} + \frac{17}{93600} a^{11} + \frac{3}{2600} a^{10} - \frac{743}{37440} a^{9} + \frac{4}{195} a^{8} - \frac{1739}{187200} a^{7} - \frac{343}{1560} a^{6} + \frac{749}{15600} a^{5} - \frac{443}{2600} a^{4} - \frac{8}{25} a^{3} + \frac{1007}{2600} a^{2} - \frac{61}{325} a - \frac{243}{650}$, $\frac{1}{97199069141497916249138664000} a^{18} + \frac{30722750332515662237}{15576773900881076321977350} a^{17} + \frac{103419129158855151122459}{97199069141497916249138664000} a^{16} - \frac{2302611640342773913363}{38571059183134093749658200} a^{15} - \frac{100291888356606216981167}{1869212868105729158637282000} a^{14} - \frac{1313912312047146117393}{8999913809397955208253580} a^{13} + \frac{1402845703442331789828917}{48599534570748958124569332000} a^{12} + \frac{768223667519951287414643}{1619984485691631937485644400} a^{11} + \frac{59301623910014230235599541}{97199069141497916249138664000} a^{10} + \frac{2558186484260452449489983}{77142118366268187499316400} a^{9} - \frac{1098125943472274576706566549}{97199069141497916249138664000} a^{8} + \frac{316870176123168231107681}{15428423673253637499863280} a^{7} + \frac{1160090055466394241431280089}{16199844856916319374856444000} a^{6} - \frac{28545504550271873269042463}{1619984485691631937485644400} a^{5} - \frac{39597186977335713965993387}{224997845234948880206339500} a^{4} - \frac{772138776001296004045179}{4736796741788397478028200} a^{3} - \frac{25443556608907239780941122}{56249461308737220051584875} a^{2} + \frac{14449918989961610301873871}{44999569046989776041267900} a - \frac{2020199504394729883847899}{5920995927235496847535250}$, $\frac{1}{52650601560829843643330357941605255334435009872000} a^{19} - \frac{168620262004900127}{50143430057933184422219388515814528889938104640} a^{18} + \frac{56774181281947925680818697371135092094192059}{52650601560829843643330357941605255334435009872000} a^{17} - \frac{8111343261690130383080141901571739689129327}{1755020052027661454777678598053508511147833662400} a^{16} - \frac{7789178171780468488101812650975291785194111}{173192768292203433037270914281596234652746743000} a^{15} - \frac{11692984977345818507549803596124934318673959}{33750385615916566438032280731798240598996801200} a^{14} - \frac{6770636487740831686098706889764745614711117693}{26325300780414921821665178970802627667217504936000} a^{13} + \frac{22723528558087417243876863399645915779020543}{58500668400922048492589286601783617038261122080} a^{12} + \frac{39518714855551408217706532137075405303184322201}{52650601560829843643330357941605255334435009872000} a^{11} + \frac{873862359985222422240670582722810202627099127}{1755020052027661454777678598053508511147833662400} a^{10} - \frac{1947226315587842478388954234114997625907312886249}{52650601560829843643330357941605255334435009872000} a^{9} - \frac{19845111078455964456341157631537034702442668111}{1755020052027661454777678598053508511147833662400} a^{8} - \frac{797752324537077935055860169931904734517475121111}{8775100260138307273888392990267542555739168312000} a^{7} + \frac{135647449722024861583088866517221200424095673157}{877510026013830727388839299026754255573916831200} a^{6} + \frac{174477474886691811534068494953214732930093263523}{731258355011525606157366082522295212978264026000} a^{5} + \frac{21308644259141713959227883935778549922652668793}{146251671002305121231473216504459042595652805200} a^{4} + \frac{50970582055902735777435224582457824812968510427}{121876392501920934359561013753715868829710671000} a^{3} + \frac{44263223187089505522580249421381112858613111}{1875021423106475913224015596211013366610933400} a^{2} - \frac{23711422226330744443813876940021664131082791091}{60938196250960467179780506876857934414855335500} a - \frac{98571765952680207736359116015986552608455397}{320727348689265616735686878299252286393975450}$

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

\(\Q(\sqrt{145}) \), 4.4.2018400.2, 5.1.2531250000.14, 10.2.657097893500976562500000000.1

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 ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R $20$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.10.19.17$x^{10} - 2 x^{4} + 4 x^{2} - 10$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
$3$3.10.9.1$x^{10} - 3$$10$$1$$9$$F_{5}\times C_2$$[\ ]_{10}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
$5$5.10.19.4$x^{10} + 105$$10$$1$$19$$F_5$$[9/4]_{4}$
5.10.19.4$x^{10} + 105$$10$$1$$19$$F_5$$[9/4]_{4}$
$29$29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
29.4.2.1$x^{4} + 145 x^{2} + 7569$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$