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Properties

Label 20.0.96363567882...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 5^{34}\cdot 11^{16}$
Root discriminant $706.64$
Ramified primes $2, 5, 11$
Class number $2180741210$ (GRH)
Class group $[2180741210]$ (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![366060061309601, 789670631905760, 924597931103020, 719764864421240, 405664528658450, 171798573324796, 55739518657820, 13871537857540, 2578759859805, 327373663100, 18802808106, -2285604420, -654705640, -57225080, 1270950, 699996, 42090, -2420, -340, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 340*x^18 - 2420*x^17 + 42090*x^16 + 699996*x^15 + 1270950*x^14 - 57225080*x^13 - 654705640*x^12 - 2285604420*x^11 + 18802808106*x^10 + 327373663100*x^9 + 2578759859805*x^8 + 13871537857540*x^7 + 55739518657820*x^6 + 171798573324796*x^5 + 405664528658450*x^4 + 719764864421240*x^3 + 924597931103020*x^2 + 789670631905760*x + 366060061309601)
 
gp: K = bnfinit(x^20 - 340*x^18 - 2420*x^17 + 42090*x^16 + 699996*x^15 + 1270950*x^14 - 57225080*x^13 - 654705640*x^12 - 2285604420*x^11 + 18802808106*x^10 + 327373663100*x^9 + 2578759859805*x^8 + 13871537857540*x^7 + 55739518657820*x^6 + 171798573324796*x^5 + 405664528658450*x^4 + 719764864421240*x^3 + 924597931103020*x^2 + 789670631905760*x + 366060061309601, 1)
 

Normalized defining polynomial

\( x^{20} - 340 x^{18} - 2420 x^{17} + 42090 x^{16} + 699996 x^{15} + 1270950 x^{14} - 57225080 x^{13} - 654705640 x^{12} - 2285604420 x^{11} + 18802808106 x^{10} + 327373663100 x^{9} + 2578759859805 x^{8} + 13871537857540 x^{7} + 55739518657820 x^{6} + 171798573324796 x^{5} + 405664528658450 x^{4} + 719764864421240 x^{3} + 924597931103020 x^{2} + 789670631905760 x + 366060061309601 \)

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:  \(963635678828500845854720000000000000000000000000000000000=2^{55}\cdot 5^{34}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $706.64$
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:  $2, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4400=2^{4}\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{4400}(1,·)$, $\chi_{4400}(2819,·)$, $\chi_{4400}(3041,·)$, $\chi_{4400}(521,·)$, $\chi_{4400}(3459,·)$, $\chi_{4400}(3281,·)$, $\chi_{4400}(339,·)$, $\chi_{4400}(1259,·)$, $\chi_{4400}(2201,·)$, $\chi_{4400}(2539,·)$, $\chi_{4400}(2721,·)$, $\chi_{4400}(1379,·)$, $\chi_{4400}(361,·)$, $\chi_{4400}(619,·)$, $\chi_{4400}(1299,·)$, $\chi_{4400}(2561,·)$, $\chi_{4400}(841,·)$, $\chi_{4400}(1081,·)$, $\chi_{4400}(3579,·)$, $\chi_{4400}(3499,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{25} a^{5} - \frac{2}{5} a^{3} - \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{1}{25}$, $\frac{1}{25} a^{6} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{25} a$, $\frac{1}{25} a^{7} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{25} a^{2} - \frac{2}{5}$, $\frac{1}{25} a^{8} + \frac{2}{5} a^{4} - \frac{1}{25} a^{3} - \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{25} a^{9} - \frac{1}{25} a^{4} - \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{625} a^{10} + \frac{1}{125} a^{8} - \frac{2}{125} a^{7} - \frac{1}{125} a^{6} - \frac{2}{625} a^{5} + \frac{3}{25} a^{4} + \frac{54}{125} a^{3} - \frac{3}{125} a^{2} + \frac{21}{125} a - \frac{24}{625}$, $\frac{1}{625} a^{11} + \frac{1}{125} a^{9} - \frac{2}{125} a^{8} - \frac{1}{125} a^{7} - \frac{2}{625} a^{6} + \frac{54}{125} a^{4} + \frac{22}{125} a^{3} - \frac{29}{125} a^{2} - \frac{149}{625} a + \frac{3}{25}$, $\frac{1}{625} a^{12} - \frac{2}{125} a^{9} - \frac{1}{125} a^{8} - \frac{2}{625} a^{7} + \frac{1}{125} a^{5} - \frac{28}{125} a^{4} + \frac{46}{125} a^{3} - \frac{149}{625} a^{2} - \frac{12}{25} a + \frac{29}{125}$, $\frac{1}{625} a^{13} - \frac{1}{125} a^{9} - \frac{2}{625} a^{8} + \frac{1}{125} a^{6} - \frac{2}{125} a^{5} + \frac{21}{125} a^{4} - \frac{24}{625} a^{3} - \frac{7}{25} a^{2} + \frac{4}{125} a + \frac{22}{125}$, $\frac{1}{3125} a^{14} + \frac{1}{3125} a^{13} + \frac{1}{3125} a^{12} + \frac{1}{3125} a^{11} + \frac{1}{3125} a^{10} - \frac{37}{3125} a^{9} - \frac{12}{3125} a^{8} - \frac{12}{3125} a^{7} + \frac{38}{3125} a^{6} + \frac{38}{3125} a^{5} + \frac{61}{3125} a^{4} + \frac{536}{3125} a^{3} + \frac{36}{3125} a^{2} + \frac{736}{3125} a + \frac{1486}{3125}$, $\frac{1}{15625} a^{15} - \frac{1}{3125} a^{13} + \frac{2}{3125} a^{12} + \frac{1}{3125} a^{11} - \frac{3}{15625} a^{10} + \frac{9}{625} a^{9} - \frac{33}{3125} a^{8} + \frac{56}{3125} a^{7} - \frac{42}{3125} a^{6} - \frac{197}{15625} a^{5} - \frac{9}{625} a^{4} - \frac{191}{3125} a^{3} + \frac{17}{3125} a^{2} - \frac{134}{3125} a - \frac{176}{15625}$, $\frac{1}{49046875} a^{16} - \frac{182}{9809375} a^{15} - \frac{1}{78475} a^{14} - \frac{1307}{9809375} a^{13} - \frac{5043}{9809375} a^{12} - \frac{35448}{49046875} a^{11} - \frac{1828}{9809375} a^{10} + \frac{13911}{1961875} a^{9} + \frac{36339}{9809375} a^{8} + \frac{113036}{9809375} a^{7} - \frac{206482}{49046875} a^{6} + \frac{57262}{9809375} a^{5} + \frac{228724}{1961875} a^{4} + \frac{3322943}{9809375} a^{3} - \frac{2561018}{9809375} a^{2} + \frac{22070429}{49046875} a + \frac{2310773}{9809375}$, $\frac{1}{49046875} a^{17} - \frac{29}{49046875} a^{15} + \frac{1086}{9809375} a^{14} + \frac{7824}{9809375} a^{13} + \frac{12077}{49046875} a^{12} + \frac{84}{1961875} a^{11} + \frac{12852}{49046875} a^{10} + \frac{2676}{228125} a^{9} + \frac{140412}{9809375} a^{8} - \frac{6209}{671875} a^{7} + \frac{548}{392375} a^{6} - \frac{3429}{671875} a^{5} - \frac{3539354}{9809375} a^{4} + \frac{1734089}{9809375} a^{3} + \frac{23682304}{49046875} a^{2} + \frac{845086}{1961875} a + \frac{16717744}{49046875}$, $\frac{1}{95647092304992852328668296875} a^{18} + \frac{966751953055636617786}{95647092304992852328668296875} a^{17} + \frac{729194365575622684564}{95647092304992852328668296875} a^{16} - \frac{2322936141113862183247108}{95647092304992852328668296875} a^{15} + \frac{700488676004508430513254}{19129418460998570465733659375} a^{14} + \frac{55948954394648530752018082}{95647092304992852328668296875} a^{13} + \frac{75989741029797112792714232}{95647092304992852328668296875} a^{12} - \frac{26589753967147116346958362}{95647092304992852328668296875} a^{11} + \frac{58312810814209340633482504}{95647092304992852328668296875} a^{10} - \frac{200048941950782552168519738}{19129418460998570465733659375} a^{9} - \frac{885085319426220906050465092}{95647092304992852328668296875} a^{8} - \frac{1585813712635605011098969622}{95647092304992852328668296875} a^{7} - \frac{1366928243549725561585592243}{95647092304992852328668296875} a^{6} - \frac{787401678783620923841408059}{95647092304992852328668296875} a^{5} - \frac{6321116095088857059099632016}{19129418460998570465733659375} a^{4} - \frac{6231047956312236911162211866}{95647092304992852328668296875} a^{3} + \frac{22850133546316821750289233854}{95647092304992852328668296875} a^{2} + \frac{46046199351953748393534298541}{95647092304992852328668296875} a - \frac{3392286006318424176704778837}{95647092304992852328668296875}$, $\frac{1}{2832206164371241898947124326091151765849895182318128984375} a^{19} - \frac{11855864760135571173026683464}{2832206164371241898947124326091151765849895182318128984375} a^{18} + \frac{15012844305196233717569064285644592123586725916061}{2832206164371241898947124326091151765849895182318128984375} a^{17} - \frac{11290201252032797683234943570445508618307627875314}{2832206164371241898947124326091151765849895182318128984375} a^{16} - \frac{223002398494916338601800761380739555822907549685658}{38797344717414272588316771590289750217121851812577109375} a^{15} + \frac{377816324095468531919549779258008988322368523903595557}{2832206164371241898947124326091151765849895182318128984375} a^{14} + \frac{31766943041548141363803572368578034025628152002338264}{65865259636540509277840100606770971298834771681816953125} a^{13} + \frac{194249380428273095911931273995873513624672887289821077}{2832206164371241898947124326091151765849895182318128984375} a^{12} - \frac{405157552169112884377333440045139404832217703785133723}{2832206164371241898947124326091151765849895182318128984375} a^{11} + \frac{545816950548145927030094962291000188323823312935480662}{2832206164371241898947124326091151765849895182318128984375} a^{10} + \frac{18582973018736079014961868662030787949468127331847343458}{2832206164371241898947124326091151765849895182318128984375} a^{9} - \frac{9735005854038620326712480023532453752176890743668624737}{2832206164371241898947124326091151765849895182318128984375} a^{8} + \frac{18054259585217633388625817515444416372847465177296123863}{2832206164371241898947124326091151765849895182318128984375} a^{7} - \frac{9225171116340468125375240545842291141123369981126805912}{2832206164371241898947124326091151765849895182318128984375} a^{6} - \frac{3862099357134812336469811826328919633129277455904262147}{2832206164371241898947124326091151765849895182318128984375} a^{5} + \frac{506326577603348391179193870680065819110091543626107725234}{2832206164371241898947124326091151765849895182318128984375} a^{4} - \frac{1160792835994286732251609225264323785412332711347836328526}{2832206164371241898947124326091151765849895182318128984375} a^{3} + \frac{886218238763975216695049759825311479610776833786326525749}{2832206164371241898947124326091151765849895182318128984375} a^{2} + \frac{563275548853846672878647993608312066047439294527744845449}{2832206164371241898947124326091151765849895182318128984375} a - \frac{59439862081832382396728252821764146003574834549412191856}{2832206164371241898947124326091151765849895182318128984375}$

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

Class group and class number

$C_{2180741210}$, which has order $2180741210$ (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:  \( 62756100051.98372 \) (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

\(\Q(\sqrt{2}) \), 4.0.51200.2, 5.5.5719140625.1, 10.10.1071794405000000000000000.2

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 R $20$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R $20$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ $20$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ $20$ $20$

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
2Data not computed
5Data not computed
11Data not computed