↑

Properties

Label 20.0.20038303066...000.29
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}$
Root discriminant $462.48$
Ramified primes $2, 3, 5, 11$
Class number $3410112640$ (GRH)
Class group $[2, 2, 2, 2, 2, 106566020]$ (GRH)
Galois group $C_2\times C_{10}$ (as 20T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![251597460606976, 0, 19285983467520, 0, 34651636166400, 0, -631791916800, 0, -231748880, 0, 164595992, 0, -20484255, 0, -62700, 0, 13750, 0, 220, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 220*x^18 + 13750*x^16 - 62700*x^14 - 20484255*x^12 + 164595992*x^10 - 231748880*x^8 - 631791916800*x^6 + 34651636166400*x^4 + 19285983467520*x^2 + 251597460606976)
 
gp: K = bnfinit(x^20 + 220*x^18 + 13750*x^16 - 62700*x^14 - 20484255*x^12 + 164595992*x^10 - 231748880*x^8 - 631791916800*x^6 + 34651636166400*x^4 + 19285983467520*x^2 + 251597460606976, 1)
 

Normalized defining polynomial

\( x^{20} + 220 x^{18} + 13750 x^{16} - 62700 x^{14} - 20484255 x^{12} + 164595992 x^{10} - 231748880 x^{8} - 631791916800 x^{6} + 34651636166400 x^{4} + 19285983467520 x^{2} + 251597460606976 \)

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:  \(200383030666749741651348266601562500000000000000000000=2^{20}\cdot 3^{10}\cdot 5^{34}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $462.48$
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:  $2, 3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3300=2^{2}\cdot 3\cdot 5^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{3300}(1,·)$, $\chi_{3300}(2371,·)$, $\chi_{3300}(2821,·)$, $\chi_{3300}(961,·)$, $\chi_{3300}(1741,·)$, $\chi_{3300}(1681,·)$, $\chi_{3300}(3299,·)$, $\chi_{3300}(1559,·)$, $\chi_{3300}(1531,·)$, $\chi_{3300}(479,·)$, $\chi_{3300}(929,·)$, $\chi_{3300}(2339,·)$, $\chi_{3300}(2791,·)$, $\chi_{3300}(389,·)$, $\chi_{3300}(1769,·)$, $\chi_{3300}(749,·)$, $\chi_{3300}(1619,·)$, $\chi_{3300}(2551,·)$, $\chi_{3300}(2911,·)$, $\chi_{3300}(509,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} + \frac{1}{8} a^{2} - \frac{1}{2}$, $\frac{1}{8} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a$, $\frac{1}{32} a^{6} - \frac{1}{16} a^{4} + \frac{1}{32} a^{2} - \frac{3}{8}$, $\frac{1}{64} a^{7} - \frac{1}{32} a^{5} + \frac{1}{64} a^{3} - \frac{3}{16} a$, $\frac{1}{256} a^{8} + \frac{1}{128} a^{6} - \frac{15}{256} a^{4} + \frac{7}{16} a^{2} - \frac{5}{16}$, $\frac{1}{256} a^{9} - \frac{1}{128} a^{7} - \frac{7}{256} a^{5} - \frac{5}{64} a^{3} + \frac{3}{8} a$, $\frac{1}{22528} a^{10} - \frac{3}{2048} a^{8} + \frac{9}{2048} a^{6} - \frac{121}{2048} a^{4} + \frac{49}{256} a^{2} + \frac{63}{128}$, $\frac{1}{45056} a^{11} + \frac{5}{4096} a^{9} + \frac{25}{4096} a^{7} + \frac{15}{4096} a^{5} - \frac{63}{512} a^{3} + \frac{87}{256} a$, $\frac{1}{45056} a^{12} - \frac{1}{45056} a^{10} + \frac{1}{4096} a^{8} + \frac{23}{4096} a^{6} + \frac{3}{64} a^{4} + \frac{51}{256} a^{2} + \frac{11}{32}$, $\frac{1}{45056} a^{13} + \frac{3}{2048} a^{9} - \frac{1}{256} a^{7} - \frac{177}{4096} a^{5} - \frac{33}{512} a^{3} + \frac{95}{256} a$, $\frac{1}{54788096} a^{14} + \frac{9}{1712128} a^{12} + \frac{9}{2490368} a^{10} - \frac{1279}{1245184} a^{8} - \frac{56653}{4980736} a^{6} - \frac{20971}{1245184} a^{4} + \frac{77185}{311296} a^{2} + \frac{18543}{77824}$, $\frac{1}{109576192} a^{15} - \frac{29}{3424256} a^{13} - \frac{509}{54788096} a^{11} - \frac{4623}{2490368} a^{9} + \frac{10227}{9961472} a^{7} + \frac{145013}{2490368} a^{5} - \frac{131967}{622592} a^{3} + \frac{65359}{155648} a$, $\frac{1}{697781190656} a^{16} - \frac{105}{15858663424} a^{14} + \frac{626691}{348890595328} a^{12} + \frac{3058713}{174445297664} a^{10} + \frac{35808195}{63434653696} a^{8} + \frac{4377901}{3964665856} a^{6} - \frac{112767323}{1982332928} a^{4} - \frac{120362027}{247791616} a^{2} - \frac{121714293}{247791616}$, $\frac{1}{1395562381312} a^{17} - \frac{105}{31717326848} a^{15} - \frac{7116797}{697781190656} a^{13} - \frac{813031}{348890595328} a^{11} + \frac{113243075}{126869307392} a^{9} - \frac{35307475}{7929331712} a^{7} - \frac{88568923}{3964665856} a^{5} + \frac{30635989}{495583232} a^{3} - \frac{81060981}{495583232} a$, $\frac{1}{4287694575555003258155310411415552} a^{18} + \frac{196847739793841811513}{535961821944375407269413801426944} a^{16} + \frac{4983012019198250781912331}{2143847287777501629077655205707776} a^{14} - \frac{9354452073939554122365778473}{1071923643888750814538827602853888} a^{12} + \frac{50655346273967603537211160625}{4287694575555003258155310411415552} a^{10} + \frac{137064739446880315785784956499}{97447603989886437685347963895808} a^{8} - \frac{158002198113634950766386855225}{12180950498735804710668495486976} a^{6} + \frac{92272492341497244170796824467}{3045237624683951177667123871744} a^{4} + \frac{620347571608370072027735960815}{1522618812341975588833561935872} a^{2} - \frac{83442714295192637982569045497}{380654703085493897208390483968}$, $\frac{1}{193212092963659556818994597759207604224} a^{19} + \frac{8074405134986106177606201}{24151511620457444602374324719900950528} a^{17} + \frac{53681631493054165469661277451}{96606046481829778409497298879603802112} a^{15} + \frac{186196856832986399876666539564503}{48303023240914889204748649439801901056} a^{13} - \frac{5040421362404944737608157896141}{17564735723969050619908599796291600384} a^{11} - \frac{8384953604288583595052369457995181}{4391183930992262654977149949072900096} a^{9} + \frac{3894860627084132709124930853319879}{548897991374032831872143743634112512} a^{7} - \frac{6843802780151703660907666494901869}{137224497843508207968035935908528128} a^{5} + \frac{1468097539542474496069899013036271}{68612248921754103984017967954264064} a^{3} - \frac{2155053458489514033705747450662393}{17153062230438525996004491988566016} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{106566020}$, which has order $3410112640$ (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:  \( 1679203762484.6501 \) (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

\(\Q(\sqrt{-165}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{11}, \sqrt{-15})\), 5.5.5719140625.4, 10.0.447641631963281250000000000.2, 10.0.39740911928558349609375.12, 10.10.368429326718750000000000.2

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 R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
3Data not computed
$5$5.10.17.27$x^{10} - 10 x^{8} + 10$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.27$x^{10} - 10 x^{8} + 10$$10$$1$$17$$C_{10}$$[2]_{2}$
11Data not computed