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Properties

Label 20.0.26166731193...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{32}\cdot 41^{18}$
Root discriminant $742.83$
Ramified primes $2, 5, 41$
Class number $1091200000$ (GRH)
Class group $[2, 2, 2, 2, 10, 10, 10, 10, 6820]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![264673853440000, 0, 368664547840000, 0, 119967187360000, 0, 17049535776000, 0, 1288069612000, 0, 56437120576, 0, 1481258865, 0, 23046100, 0, 199670, 0, 820, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 820*x^18 + 199670*x^16 + 23046100*x^14 + 1481258865*x^12 + 56437120576*x^10 + 1288069612000*x^8 + 17049535776000*x^6 + 119967187360000*x^4 + 368664547840000*x^2 + 264673853440000)
 
gp: K = bnfinit(x^20 + 820*x^18 + 199670*x^16 + 23046100*x^14 + 1481258865*x^12 + 56437120576*x^10 + 1288069612000*x^8 + 17049535776000*x^6 + 119967187360000*x^4 + 368664547840000*x^2 + 264673853440000, 1)
 

Normalized defining polynomial

\( x^{20} + 820 x^{18} + 199670 x^{16} + 23046100 x^{14} + 1481258865 x^{12} + 56437120576 x^{10} + 1288069612000 x^{8} + 17049535776000 x^{6} + 119967187360000 x^{4} + 368664547840000 x^{2} + 264673853440000 \)

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:  \(2616673119324506454012530994165039062500000000000000000000=2^{20}\cdot 5^{32}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $742.83$
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:  $2, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4100=2^{2}\cdot 5^{2}\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4100}(1,·)$, $\chi_{4100}(2051,·)$, $\chi_{4100}(901,·)$, $\chi_{4100}(2951,·)$, $\chi_{4100}(461,·)$, $\chi_{4100}(2511,·)$, $\chi_{4100}(1171,·)$, $\chi_{4100}(3221,·)$, $\chi_{4100}(1371,·)$, $\chi_{4100}(3421,·)$, $\chi_{4100}(291,·)$, $\chi_{4100}(2341,·)$, $\chi_{4100}(681,·)$, $\chi_{4100}(2731,·)$, $\chi_{4100}(1261,·)$, $\chi_{4100}(3311,·)$, $\chi_{4100}(1841,·)$, $\chi_{4100}(3891,·)$, $\chi_{4100}(631,·)$, $\chi_{4100}(2681,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{7} + \frac{3}{16} a^{3} + \frac{1}{4} a$, $\frac{1}{64} a^{8} - \frac{1}{32} a^{6} + \frac{1}{64} a^{4} + \frac{1}{16} a^{2}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} + \frac{1}{128} a^{5} + \frac{1}{32} a^{3}$, $\frac{1}{5248} a^{10} + \frac{5}{128} a^{6} + \frac{3}{64} a^{4} - \frac{3}{16} a^{2}$, $\frac{1}{10496} a^{11} - \frac{1}{256} a^{9} + \frac{7}{256} a^{7} - \frac{11}{256} a^{5} + \frac{5}{64} a^{3} + \frac{1}{4} a$, $\frac{1}{482816} a^{12} - \frac{45}{482816} a^{10} - \frac{17}{11776} a^{8} - \frac{687}{11776} a^{6} + \frac{73}{2944} a^{4} + \frac{57}{184} a^{2} + \frac{3}{23}$, $\frac{1}{4828160} a^{13} - \frac{9}{965632} a^{11} + \frac{15}{23552} a^{9} + \frac{709}{23552} a^{7} - \frac{1}{46} a^{5} + \frac{1631}{7360} a^{3} - \frac{31}{92} a$, $\frac{1}{4828160} a^{14} - \frac{1}{965632} a^{12} + \frac{71}{965632} a^{10} - \frac{163}{23552} a^{8} + \frac{27}{1472} a^{6} - \frac{27}{3680} a^{4} + \frac{171}{368} a^{2} - \frac{11}{23}$, $\frac{1}{48281600} a^{15} + \frac{197}{4828160} a^{11} - \frac{1}{368} a^{9} + \frac{1033}{235520} a^{7} + \frac{1167}{147200} a^{5} - \frac{2239}{14720} a^{3} - \frac{3}{184} a$, $\frac{1}{101198233600} a^{16} + \frac{51}{1011982336} a^{14} - \frac{10073}{10119823360} a^{12} - \frac{68159}{1011982336} a^{10} - \frac{777547}{493649920} a^{8} + \frac{4340367}{308531200} a^{6} - \frac{2282139}{30853120} a^{4} - \frac{141691}{385664} a^{2} + \frac{2327}{6026}$, $\frac{1}{6274290483200} a^{17} + \frac{7563}{1568572620800} a^{15} - \frac{64569}{627429048320} a^{13} + \frac{13004437}{313714524160} a^{11} - \frac{117608587}{30606295040} a^{9} - \frac{331784673}{19128934400} a^{7} - \frac{1136103}{9564467200} a^{5} - \frac{127019}{23911168} a^{3} + \frac{114201}{373612} a$, $\frac{1}{6743403223009138121471688704000} a^{18} + \frac{116793195807585521}{84292540287614226518396108800} a^{16} + \frac{66533995193894401870047}{674340322300913812147168870400} a^{14} + \frac{14350150658545711736903}{67434032230091381214716887040} a^{12} + \frac{470746054475006208282737}{46506229124200952561873715200} a^{10} + \frac{249551577421782893813581759}{41118312335421573911412736000} a^{8} - \frac{57432982415946944354806213}{2055915616771078695570636800} a^{6} - \frac{4018675769251850987344179}{102795780838553934778531840} a^{4} - \frac{155173646405500750740327}{1284947260481924184731648} a^{2} - \frac{274715281529542016661}{647654869194518238272}$, $\frac{1}{552959064286749325960678473728000} a^{19} + \frac{2193268521924833}{168585080575228453036792217600} a^{17} - \frac{12843295996831255203129}{1348680644601827624294337740800} a^{15} - \frac{10553061291983684400593}{134868064460182762429433774080} a^{13} + \frac{267727657179828832249097}{93012458248401905123747430400} a^{11} + \frac{5679157540244433439541555359}{3371701611504569060735844352000} a^{9} - \frac{44218038837364540720744701}{4111831233542157391141273600} a^{7} - \frac{33863881862909503705705863}{1027957808385539347785318400} a^{5} - \frac{2287377701583069691099371}{12849472604819241847316480} a^{3} + \frac{12373906875889134324877}{40154601890060130772864} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{10}\times C_{10}\times C_{10}\times C_{10}\times C_{6820}$, which has order $1091200000$ (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:  \( \frac{7509509647}{192763837804946233753600} a^{19} + \frac{8905091871}{293847313727052185600} a^{17} + \frac{15281196866509}{2350778509816417484800} a^{15} + \frac{146196929192229}{235077850981641748480} a^{13} + \frac{1001295550254727}{32424531169881620480} a^{11} + \frac{958447404763948833}{1175389254908208742400} a^{9} + \frac{74233942829452161}{7167007651879321600} a^{7} + \frac{63351950485606467}{1791751912969830400} a^{5} - \frac{1326160168480165}{4479379782424576} a^{3} - \frac{92505961003365}{69990309100384} a \) (order $4$)
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:  \( 523576436698285.7 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{10}$ (as 20T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 20
The 20 conjugacy class representatives for $C_2\times C_{10}$
Character table for $C_2\times C_{10}$

Intermediate fields

\(\Q(\sqrt{-41}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{41}) \), \(\Q(i, \sqrt{41})\), 5.5.1103812890625.4, 10.0.51153427249056406250000000000.1, 10.0.1247644567050156250000000000.4, 10.10.49954518797906646728515625.2

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$5$5.5.8.3$x^{5} - 5 x^{4} + 30$$5$$1$$8$$C_5$$[2]$
5.5.8.3$x^{5} - 5 x^{4} + 30$$5$$1$$8$$C_5$$[2]$
5.5.8.3$x^{5} - 5 x^{4} + 30$$5$$1$$8$$C_5$$[2]$
5.5.8.3$x^{5} - 5 x^{4} + 30$$5$$1$$8$$C_5$$[2]$
$41$41.10.9.4$x^{10} - 1912896$$10$$1$$9$$C_{10}$$[\ ]_{10}$
41.10.9.4$x^{10} - 1912896$$10$$1$$9$$C_{10}$$[\ ]_{10}$