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Properties

Label 20.0.39848390734...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{33}\cdot 3^{17}\cdot 5^{38}\cdot 11^{5}\cdot 19^{10}$
Root discriminant $1349.03$
Ramified primes $2, 3, 5, 11, 19$
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![1312243974648576, 249655655692800, -351854399957760, -43566888614400, 55180067339520, 1346584988160, -3979561633920, 72940867200, 219025145760, -4850582400, -8022023552, 85989600, 206810960, -334800, -3692720, -2160, 43720, 0, -310, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 310*x^18 + 43720*x^16 - 2160*x^15 - 3692720*x^14 - 334800*x^13 + 206810960*x^12 + 85989600*x^11 - 8022023552*x^10 - 4850582400*x^9 + 219025145760*x^8 + 72940867200*x^7 - 3979561633920*x^6 + 1346584988160*x^5 + 55180067339520*x^4 - 43566888614400*x^3 - 351854399957760*x^2 + 249655655692800*x + 1312243974648576)
 
gp: K = bnfinit(x^20 - 310*x^18 + 43720*x^16 - 2160*x^15 - 3692720*x^14 - 334800*x^13 + 206810960*x^12 + 85989600*x^11 - 8022023552*x^10 - 4850582400*x^9 + 219025145760*x^8 + 72940867200*x^7 - 3979561633920*x^6 + 1346584988160*x^5 + 55180067339520*x^4 - 43566888614400*x^3 - 351854399957760*x^2 + 249655655692800*x + 1312243974648576, 1)
 

Normalized defining polynomial

\( x^{20} - 310 x^{18} + 43720 x^{16} - 2160 x^{15} - 3692720 x^{14} - 334800 x^{13} + 206810960 x^{12} + 85989600 x^{11} - 8022023552 x^{10} - 4850582400 x^{9} + 219025145760 x^{8} + 72940867200 x^{7} - 3979561633920 x^{6} + 1346584988160 x^{5} + 55180067339520 x^{4} - 43566888614400 x^{3} - 351854399957760 x^{2} + 249655655692800 x + 1312243974648576 \)

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:  \(398483907346819732159008478125000000000000000000000000000000000=2^{33}\cdot 3^{17}\cdot 5^{38}\cdot 11^{5}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1349.03$
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:  $2, 3, 5, 11, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{12} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{7} - \frac{1}{12} a^{5} - \frac{1}{6} a^{3}$, $\frac{1}{24} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{24} a^{9} + \frac{1}{6} a^{3}$, $\frac{1}{1200} a^{10} + \frac{1}{60} a^{8} - \frac{1}{30} a^{6} + \frac{1}{10} a^{5} - \frac{2}{15} a^{4} - \frac{4}{15} a^{2} - \frac{12}{25}$, $\frac{1}{1200} a^{11} + \frac{1}{60} a^{9} - \frac{1}{30} a^{7} + \frac{1}{60} a^{6} + \frac{7}{60} a^{5} - \frac{1}{6} a^{4} + \frac{7}{30} a^{3} - \frac{1}{3} a^{2} - \frac{12}{25} a$, $\frac{1}{7200} a^{12} - \frac{1}{3600} a^{10} + \frac{1}{60} a^{8} + \frac{1}{60} a^{7} - \frac{1}{90} a^{6} - \frac{7}{60} a^{5} + \frac{2}{9} a^{4} + \frac{1}{6} a^{3} + \frac{43}{150} a^{2} - \frac{6}{25}$, $\frac{1}{7200} a^{13} - \frac{1}{3600} a^{11} + \frac{1}{60} a^{9} + \frac{1}{60} a^{8} - \frac{1}{90} a^{7} - \frac{1}{30} a^{6} - \frac{1}{36} a^{5} - \frac{1}{6} a^{4} - \frac{16}{75} a^{3} + \frac{1}{3} a^{2} - \frac{6}{25} a$, $\frac{1}{7200} a^{14} + \frac{1}{3600} a^{10} + \frac{1}{60} a^{9} - \frac{1}{360} a^{8} - \frac{1}{20} a^{5} + \frac{22}{225} a^{4} + \frac{1}{6} a^{3} - \frac{4}{15} a^{2} - \frac{9}{25}$, $\frac{1}{57600} a^{15} - \frac{1}{14400} a^{14} + \frac{1}{28800} a^{13} - \frac{1}{7200} a^{11} - \frac{1}{2880} a^{10} - \frac{1}{720} a^{9} + \frac{17}{1440} a^{8} + \frac{1}{720} a^{7} + \frac{1}{30} a^{6} - \frac{371}{3600} a^{5} - \frac{7}{450} a^{4} - \frac{4}{75} a^{3} + \frac{17}{60} a^{2} - \frac{21}{50} a - \frac{1}{4}$, $\frac{1}{172800} a^{16} + \frac{1}{86400} a^{14} - \frac{1}{21600} a^{12} - \frac{1}{4800} a^{11} - \frac{1}{5400} a^{10} - \frac{1}{160} a^{9} + \frac{43}{2160} a^{8} + \frac{1}{60} a^{7} - \frac{311}{10800} a^{6} - \frac{1}{10} a^{5} - \frac{103}{450} a^{4} - \frac{11}{60} a^{3} + \frac{49}{150} a^{2} - \frac{23}{100} a + \frac{11}{25}$, $\frac{1}{345600} a^{17} + \frac{1}{172800} a^{15} - \frac{1}{43200} a^{13} + \frac{1}{28800} a^{12} - \frac{1}{10800} a^{11} - \frac{1}{14400} a^{10} + \frac{43}{4320} a^{9} + \frac{1}{120} a^{8} - \frac{851}{21600} a^{7} + \frac{1}{72} a^{6} + \frac{77}{900} a^{5} - \frac{17}{72} a^{4} + \frac{49}{300} a^{3} - \frac{79}{200} a^{2} + \frac{11}{50} a - \frac{4}{25}$, $\frac{1}{11802464527380938455978177344000} a^{18} + \frac{29290356240912247506079}{131138494748677093955313081600} a^{17} + \frac{1362019410962824851531119}{2950616131845234613994544336000} a^{16} - \frac{279748204432401583153043}{32784623687169273488828270400} a^{15} + \frac{178782554174697980762421619}{2950616131845234613994544336000} a^{14} - \frac{4131939238444174634275163}{65569247374338546977656540800} a^{13} - \frac{24576938975159896710123163}{1475308065922617306997272168000} a^{12} - \frac{80410797262742770403599}{273205197393077279073568920} a^{11} - \frac{14526162324800154325027039}{737654032961308653498636084000} a^{10} - \frac{261308927245276642391799349}{16392311843584636744414135200} a^{9} + \frac{14408755785211816224581175889}{737654032961308653498636084000} a^{8} - \frac{326256148609002422291156881}{8196155921792318372207067600} a^{7} - \frac{651307856917806731645538827}{40980779608961591861035338000} a^{6} - \frac{3284163389907482455232801}{102451949022403979652588345} a^{5} - \frac{165330265996926246710758822}{2561298725560099491314708625} a^{4} + \frac{52812128166067329407568539}{1366025986965386395367844600} a^{3} + \frac{519747318995297511731351153}{1138354989137821996139870500} a^{2} - \frac{85784170967691498930842497}{227670997827564399227974100} a + \frac{99921680993696840305337909}{569177494568910998069935250}$, $\frac{1}{5785391450895013735797428327951194773123568398695151043135296000} a^{19} - \frac{23339509117737227478490421664863}{1157078290179002747159485665590238954624713679739030208627059200} a^{18} - \frac{6171888925628365542129794882911178163385792150496970216649}{5785391450895013735797428327951194773123568398695151043135296000} a^{17} - \frac{167478344498523688829996788342417915228022088507066648349}{289269572544750686789871416397559738656178419934757552156764800} a^{16} + \frac{3241474638488899721359874108199218953193703759734065049627}{723173931361876716974678540993899346640446049836893880391912000} a^{15} - \frac{67588932654604450728811850387156761269702757215630356327}{1807934828404691792436696352484748366601115124592234700979780} a^{14} + \frac{21874207985491970317483992560891757975404397685057785058681}{361586965680938358487339270496949673320223024918446940195956000} a^{13} + \frac{18988458377860811658348931531501083675533436027086389221001}{289269572544750686789871416397559738656178419934757552156764800} a^{12} - \frac{33769823582742604304766286447839674996549055301323240950911}{90396741420234589621834817624237418330055756229611735048989000} a^{11} - \frac{971541684840174954553873886790624389680096605866582966947}{7231739313618767169746785409938993466404460498368938803919120} a^{10} - \frac{3763510885678699889620867925040491640201366005718063801962561}{361586965680938358487339270496949673320223024918446940195956000} a^{9} + \frac{577755175257669598860076151153861974456299801476784773355791}{36158696568093835848733927049694967332022302491844694019595600} a^{8} - \frac{2118428818307505856435675621431601842792630383977717762471487}{120528988560312786162446423498983224440074341639482313398652000} a^{7} - \frac{3971406290977992890203839112073335513564158882058077496697}{482115954241251144649785693995932897760297366557929253594608} a^{6} + \frac{626375925901420617870913602667707553945525383858812771391731}{6696054920017377009024801305499068024448574535526795188814000} a^{5} + \frac{200807354892945626072413887932733201993873373153376323026307}{1004408238002606551353720195824860203667286180329019278322100} a^{4} - \frac{215480803643268076043311702576589121308450277999579159930307}{3348027460008688504512400652749534012224287267763397594407000} a^{3} + \frac{46341917137141832514933759966181667885068075603773103439947}{133921098400347540180496026109981360488971490710535903776280} a^{2} + \frac{262993425571816898588278673966955604229271044483799106026003}{558004576668114750752066775458255668704047877960566265734500} a + \frac{41960318438200705321143273637939409389484497016386061518013}{111600915333622950150413355091651133740809575592113253146900}$

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

$D_4\times F_5$ (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

\(\Q(\sqrt{-95}) \), 4.0.9530400.6, 5.1.2531250000.13, 10.0.79324636420898437500000000.1

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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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$2.10.14.1$x^{10} - 2 x^{6} + 2 x^{5} + 2 x^{2} + 2$$10$$1$$14$$F_{5}\times C_2$$[2]_{5}^{4}$
2.10.19.33$x^{10} - 6 x^{4} + 4 x^{2} - 14$$10$$1$$19$$F_{5}\times C_2$$[3]_{5}^{4}$
$3$3.10.9.2$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$5.10.19.3$x^{10} + 30$$10$$1$$19$$F_5$$[9/4]_{4}$
5.10.19.3$x^{10} + 30$$10$$1$$19$$F_5$$[9/4]_{4}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2}$
11.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$19$19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
19.4.2.1$x^{4} + 57 x^{2} + 1444$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$