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Properties

Label 20.0.24432145751...4641.4
Degree $20$
Signature $[0, 10]$
Discriminant $11^{18}\cdot 41^{19}$
Root discriminant $294.71$
Ramified primes $11, 41$
Class number $3065931370$ (GRH)
Class group $[3065931370]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![214582951936, 57786515456, 108451571712, -7922589696, 33352186240, 1033992256, 8285369632, -122399344, 728068088, -67318932, 21164846, -6579077, 1358797, -96856, 17524, -7106, 1506, 100, 22, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 22*x^18 + 100*x^17 + 1506*x^16 - 7106*x^15 + 17524*x^14 - 96856*x^13 + 1358797*x^12 - 6579077*x^11 + 21164846*x^10 - 67318932*x^9 + 728068088*x^8 - 122399344*x^7 + 8285369632*x^6 + 1033992256*x^5 + 33352186240*x^4 - 7922589696*x^3 + 108451571712*x^2 + 57786515456*x + 214582951936)
 
gp: K = bnfinit(x^20 - x^19 + 22*x^18 + 100*x^17 + 1506*x^16 - 7106*x^15 + 17524*x^14 - 96856*x^13 + 1358797*x^12 - 6579077*x^11 + 21164846*x^10 - 67318932*x^9 + 728068088*x^8 - 122399344*x^7 + 8285369632*x^6 + 1033992256*x^5 + 33352186240*x^4 - 7922589696*x^3 + 108451571712*x^2 + 57786515456*x + 214582951936, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 22 x^{18} + 100 x^{17} + 1506 x^{16} - 7106 x^{15} + 17524 x^{14} - 96856 x^{13} + 1358797 x^{12} - 6579077 x^{11} + 21164846 x^{10} - 67318932 x^{9} + 728068088 x^{8} - 122399344 x^{7} + 8285369632 x^{6} + 1033992256 x^{5} + 33352186240 x^{4} - 7922589696 x^{3} + 108451571712 x^{2} + 57786515456 x + 214582951936 \)

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:  \(24432145751046714758355457553172804266229073614641=11^{18}\cdot 41^{19}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $294.71$
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:  $11, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(451=11\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{451}(256,·)$, $\chi_{451}(1,·)$, $\chi_{451}(2,·)$, $\chi_{451}(4,·)$, $\chi_{451}(8,·)$, $\chi_{451}(74,·)$, $\chi_{451}(141,·)$, $\chi_{451}(16,·)$, $\chi_{451}(148,·)$, $\chi_{451}(282,·)$, $\chi_{451}(32,·)$, $\chi_{451}(226,·)$, $\chi_{451}(37,·)$, $\chi_{451}(296,·)$, $\chi_{451}(64,·)$, $\chi_{451}(113,·)$, $\chi_{451}(128,·)$, $\chi_{451}(244,·)$, $\chi_{451}(122,·)$, $\chi_{451}(61,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{5} - \frac{1}{16} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{64} a^{7} - \frac{1}{64} a^{6} - \frac{1}{64} a^{5} - \frac{3}{64} a^{4} + \frac{1}{16} a^{2}$, $\frac{1}{128} a^{8} - \frac{1}{64} a^{6} - \frac{1}{32} a^{5} + \frac{5}{128} a^{4} + \frac{5}{32} a^{3} + \frac{7}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{256} a^{9} - \frac{1}{128} a^{7} - \frac{1}{64} a^{6} + \frac{5}{256} a^{5} - \frac{3}{64} a^{4} - \frac{9}{64} a^{3} + \frac{3}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{512} a^{10} - \frac{1}{256} a^{8} - \frac{15}{512} a^{6} - \frac{1}{32} a^{5} + \frac{3}{32} a^{3} + \frac{7}{32} a^{2} + \frac{1}{8} a$, $\frac{1}{1024} a^{11} - \frac{1}{1024} a^{10} - \frac{1}{512} a^{8} + \frac{5}{1024} a^{7} + \frac{7}{1024} a^{6} + \frac{9}{512} a^{5} + \frac{7}{256} a^{4} + \frac{5}{128} a^{3} + \frac{1}{32} a^{2}$, $\frac{1}{2048} a^{12} - \frac{1}{2048} a^{11} - \frac{1}{1024} a^{10} + \frac{1}{1024} a^{9} + \frac{1}{2048} a^{8} - \frac{1}{2048} a^{7} - \frac{1}{128} a^{6} - \frac{1}{128} a^{5} - \frac{7}{128} a^{4} - \frac{15}{128} a^{3} - \frac{3}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{8192} a^{13} - \frac{1}{8192} a^{12} + \frac{1}{4096} a^{11} - \frac{3}{4096} a^{10} - \frac{7}{8192} a^{9} + \frac{15}{8192} a^{8} - \frac{11}{2048} a^{7} + \frac{17}{1024} a^{6} - \frac{1}{512} a^{5} - \frac{1}{512} a^{4} + \frac{27}{128} a^{3} + \frac{3}{16} a^{2} - \frac{1}{2} a$, $\frac{1}{32768} a^{14} + \frac{5}{32768} a^{12} - \frac{1}{4096} a^{11} - \frac{5}{32768} a^{10} - \frac{1}{2048} a^{9} - \frac{57}{32768} a^{8} + \frac{19}{4096} a^{7} - \frac{7}{4096} a^{6} + \frac{1}{64} a^{5} - \frac{25}{2048} a^{4} + \frac{57}{256} a^{3} + \frac{25}{128} a^{2} + \frac{7}{16} a$, $\frac{1}{32768} a^{15} + \frac{1}{32768} a^{13} - \frac{1}{8192} a^{12} - \frac{13}{32768} a^{11} + \frac{1}{4096} a^{10} - \frac{29}{32768} a^{9} + \frac{23}{8192} a^{8} + \frac{15}{4096} a^{7} - \frac{1}{1024} a^{6} - \frac{21}{2048} a^{5} - \frac{13}{512} a^{4} - \frac{1}{64} a^{3}$, $\frac{1}{524288} a^{16} - \frac{3}{524288} a^{15} - \frac{1}{524288} a^{14} - \frac{31}{524288} a^{13} + \frac{29}{524288} a^{12} + \frac{255}{524288} a^{11} + \frac{133}{524288} a^{10} + \frac{27}{524288} a^{9} + \frac{1023}{262144} a^{8} + \frac{427}{65536} a^{7} - \frac{9}{512} a^{6} - \frac{901}{32768} a^{5} + \frac{235}{16384} a^{4} + \frac{63}{1024} a^{3} + \frac{209}{1024} a^{2} - \frac{59}{128} a - \frac{3}{8}$, $\frac{1}{17536385024} a^{17} + \frac{1213}{2192048128} a^{16} - \frac{42185}{8768192512} a^{15} + \frac{56643}{8768192512} a^{14} - \frac{55597}{2192048128} a^{13} + \frac{787575}{8768192512} a^{12} - \frac{1315051}{8768192512} a^{11} + \frac{1562033}{8768192512} a^{10} + \frac{22657303}{17536385024} a^{9} + \frac{18001017}{8768192512} a^{8} + \frac{14872025}{2192048128} a^{7} - \frac{22663165}{1096024064} a^{6} + \frac{10986127}{1096024064} a^{5} + \frac{3520353}{548012032} a^{4} - \frac{1630983}{17125376} a^{3} + \frac{4983947}{34250752} a^{2} - \frac{1224225}{4281344} a + \frac{46967}{267584}$, $\frac{1}{25708340445184} a^{18} + \frac{59}{12854170222592} a^{17} + \frac{959427}{12854170222592} a^{16} + \frac{186444713}{12854170222592} a^{15} - \frac{89451647}{6427085111296} a^{14} + \frac{349837267}{12854170222592} a^{13} - \frac{268359237}{12854170222592} a^{12} - \frac{3939636573}{12854170222592} a^{11} - \frac{3409841157}{25708340445184} a^{10} - \frac{1181269397}{6427085111296} a^{9} - \frac{225440083}{6427085111296} a^{8} + \frac{3747756503}{803385638912} a^{7} + \frac{19823499001}{1606771277824} a^{6} - \frac{2878884925}{401692819456} a^{5} - \frac{25081949997}{401692819456} a^{4} - \frac{188895981}{1357070336} a^{3} - \frac{5543632931}{25105801216} a^{2} - \frac{270207179}{3138225152} a - \frac{12968891}{196139072}$, $\frac{1}{212316315303037356477498526049342703064757649276928} a^{19} + \frac{2393198402032134040038238348944076265}{212316315303037356477498526049342703064757649276928} a^{18} - \frac{262878204466299975971187717157042212107}{13269769706439834779843657878083918941547353079808} a^{17} - \frac{37720459454497634215635771441495297416381375}{53079078825759339119374631512335675766189412319232} a^{16} + \frac{42552157859926118120998543958232467134502917}{106158157651518678238749263024671351532378824638464} a^{15} + \frac{508180906269210766623194440018507761843266417}{106158157651518678238749263024671351532378824638464} a^{14} + \frac{1042703548477336156941083892836981141216683425}{26539539412879669559687315756167837883094706159616} a^{13} - \frac{3861745327981715331981783769188149964523672337}{26539539412879669559687315756167837883094706159616} a^{12} + \frac{35605738175384953999742112981140211318669796349}{212316315303037356477498526049342703064757649276928} a^{11} - \frac{2062146098253609441226349191252843094615944959}{3480595332836677975040959443431847591225535234048} a^{10} + \frac{3228419321051245077182818794211692020806550571}{13269769706439834779843657878083918941547353079808} a^{9} - \frac{206047853720077973100411013977440157390403578605}{53079078825759339119374631512335675766189412319232} a^{8} + \frac{82279426816382323611034850964010555725429409991}{13269769706439834779843657878083918941547353079808} a^{7} + \frac{398084500735978164453806020068465335916244731967}{13269769706439834779843657878083918941547353079808} a^{6} + \frac{10596702134974190001415429271540116051863471589}{1658721213304979347480457234760489867693419134976} a^{5} + \frac{51432012456717818904036285852510910121552832901}{3317442426609958694960914469520979735386838269952} a^{4} + \frac{5716825549928052556107515352662342260586712919}{414680303326244836870114308690122466923354783744} a^{3} + \frac{41569100922505642193766195081504330647868077531}{207340151663122418435057154345061233461677391872} a^{2} + \frac{3187980298097172087511848531808331652533548251}{25917518957890302304382144293132654182709673984} a + \frac{19446568939307699087805744609609655262413311}{43779592834274159297942811305967321254577152}$

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

Class group and class number

$C_{3065931370}$, which has order $3065931370$ (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:  $9$
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:  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:  \( 9382131257586.73 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{41}) \), 4.0.8339441.1, 5.5.41371966801.4, 10.10.70177225116304893117641.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.1.0.1}{1} }^{20}$ $20$ ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ $20$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ $20$ $20$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{20}$ R ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
11Data not computed
41Data not computed