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Properties

Label 20.0.67955065915...000.18
Degree $20$
Signature $[0, 10]$
Discriminant $2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}$
Root discriminant $438.14$
Ramified primes $2, 3, 5, 41$
Class number $8260977664$ (GRH)
Class group $[2, 2, 2, 2, 2, 2, 129077776]$ (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![94028983151082849, 0, 9481858962789978, 0, 481702575344139, 0, 16672949786898, 0, 418017577635, 0, 7950176568, 0, 117893880, 0, 1231902, 0, 12825, 0, 54, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 54*x^18 + 12825*x^16 + 1231902*x^14 + 117893880*x^12 + 7950176568*x^10 + 418017577635*x^8 + 16672949786898*x^6 + 481702575344139*x^4 + 9481858962789978*x^2 + 94028983151082849)
 
gp: K = bnfinit(x^20 + 54*x^18 + 12825*x^16 + 1231902*x^14 + 117893880*x^12 + 7950176568*x^10 + 418017577635*x^8 + 16672949786898*x^6 + 481702575344139*x^4 + 9481858962789978*x^2 + 94028983151082849, 1)
 

Normalized defining polynomial

\( x^{20} + 54 x^{18} + 12825 x^{16} + 1231902 x^{14} + 117893880 x^{12} + 7950176568 x^{10} + 418017577635 x^{8} + 16672949786898 x^{6} + 481702575344139 x^{4} + 9481858962789978 x^{2} + 94028983151082849 \)

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:  \(67955065915961530473358856164812748971048960000000000=2^{40}\cdot 3^{10}\cdot 5^{10}\cdot 41^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $438.14$
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:  $2, 3, 5, 41$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4920=2^{3}\cdot 3\cdot 5\cdot 41\)
Dirichlet character group:    $\lbrace$$\chi_{4920}(1,·)$, $\chi_{4920}(3139,·)$, $\chi_{4920}(1349,·)$, $\chi_{4920}(961,·)$, $\chi_{4920}(2189,·)$, $\chi_{4920}(4561,·)$, $\chi_{4920}(4099,·)$, $\chi_{4920}(1991,·)$, $\chi_{4920}(3481,·)$, $\chi_{4920}(4699,·)$, $\chi_{4920}(3551,·)$, $\chi_{4920}(4321,·)$, $\chi_{4920}(619,·)$, $\chi_{4920}(3749,·)$, $\chi_{4920}(4391,·)$, $\chi_{4920}(4459,·)$, $\chi_{4920}(3311,·)$, $\chi_{4920}(2789,·)$, $\chi_{4920}(2951,·)$, $\chi_{4920}(2429,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{27} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{27} a^{5} - \frac{1}{3} a$, $\frac{1}{243} a^{6} - \frac{1}{81} a^{4} - \frac{2}{27} a^{2}$, $\frac{1}{243} a^{7} - \frac{1}{81} a^{5} + \frac{1}{27} a^{3} + \frac{1}{3} a$, $\frac{1}{729} a^{8} + \frac{1}{27} a^{2}$, $\frac{1}{6561} a^{9} + \frac{1}{729} a^{7} + \frac{1}{243} a^{5} + \frac{10}{243} a^{3} + \frac{1}{9} a$, $\frac{1}{39366} a^{10} + \frac{1}{4374} a^{8} - \frac{1}{729} a^{6} + \frac{19}{1458} a^{4} + \frac{1}{18} a^{2} - \frac{1}{2}$, $\frac{1}{39366} a^{11} - \frac{1}{13122} a^{9} - \frac{11}{1458} a^{5} + \frac{5}{486} a^{3} - \frac{7}{18} a$, $\frac{1}{354294} a^{12} + \frac{1}{118098} a^{10} + \frac{4}{6561} a^{8} - \frac{23}{13122} a^{6} - \frac{65}{4374} a^{4} + \frac{25}{162} a^{2}$, $\frac{1}{1062882} a^{13} - \frac{1}{177147} a^{11} - \frac{1}{39366} a^{9} - \frac{59}{39366} a^{7} - \frac{88}{6561} a^{5} - \frac{2}{243} a^{3} + \frac{1}{18} a$, $\frac{1}{3188646} a^{14} + \frac{1}{1062882} a^{12} + \frac{13}{118098} a^{8} + \frac{79}{39366} a^{6} - \frac{23}{4374} a^{4} + \frac{11}{81} a^{2}$, $\frac{1}{6377292} a^{15} - \frac{1}{6377292} a^{14} - \frac{1}{2125764} a^{13} + \frac{1}{1062882} a^{12} - \frac{5}{708588} a^{11} - \frac{1}{118098} a^{10} + \frac{5}{118098} a^{9} - \frac{65}{118098} a^{8} - \frac{19}{78732} a^{7} + \frac{17}{19683} a^{6} + \frac{1}{6561} a^{5} + \frac{1}{972} a^{4} - \frac{23}{972} a^{3} + \frac{4}{27} a^{2} - \frac{1}{12} a - \frac{1}{4}$, $\frac{1}{4189880844} a^{16} - \frac{13}{349156737} a^{14} - \frac{1}{2125764} a^{13} + \frac{241}{465542316} a^{12} - \frac{7}{708588} a^{11} - \frac{139}{77590386} a^{10} - \frac{1}{39366} a^{9} + \frac{26561}{51726924} a^{8} - \frac{157}{78732} a^{7} - \frac{905}{4310577} a^{6} + \frac{383}{26244} a^{5} + \frac{7}{53217} a^{4} - \frac{13}{324} a^{3} - \frac{1069}{23652} a^{2} - \frac{1}{18} a + \frac{85}{292}$, $\frac{1}{4189880844} a^{17} - \frac{13}{349156737} a^{15} - \frac{1}{6377292} a^{14} - \frac{197}{465542316} a^{13} - \frac{1}{2125764} a^{12} + \frac{299}{77590386} a^{11} - \frac{3661}{51726924} a^{9} - \frac{13}{236196} a^{8} - \frac{715}{8621154} a^{7} - \frac{79}{78732} a^{6} - \frac{7310}{478953} a^{5} - \frac{139}{8748} a^{4} - \frac{3791}{70956} a^{3} - \frac{10}{81} a^{2} + \frac{401}{876} a$, $\frac{1}{507471299864612090451720605606616396} a^{18} - \frac{962543499189522302464003}{9397616664159483156513344548270674} a^{16} - \frac{535083291247430988500069255}{4698808332079741578256672274135337} a^{14} - \frac{5123162798782954004304192143}{9397616664159483156513344548270674} a^{12} - \frac{1}{78732} a^{11} + \frac{4141926239390350180765844065}{696119752900702456038025522094124} a^{10} - \frac{1}{26244} a^{9} - \frac{24482422426489657278762319279}{348059876450351228019012761047062} a^{8} + \frac{1}{729} a^{7} + \frac{130414929497858506420706083462}{174029938225175614009506380523531} a^{6} - \frac{13}{2916} a^{5} - \frac{38435165892010490383193808457}{6445553267599096815166902982353} a^{4} - \frac{7}{972} a^{3} + \frac{99190114800238893783235754323}{954896780385051380024726367756} a^{2} + \frac{11}{36} a - \frac{260135619307245635514787541}{3929616380185396625616157892}$, $\frac{1}{7905895380590791757147355314745476833284} a^{19} - \frac{4065154827115592881672599905}{146405470010940588095321394717508830246} a^{17} - \frac{4628950299632489622295564175981}{73202735005470294047660697358754415123} a^{15} + \frac{2066320883611675893354565743962}{73202735005470294047660697358754415123} a^{13} - \frac{1}{708588} a^{12} - \frac{56185281202947478791424171080031}{10844849630440043562616399608704357796} a^{11} - \frac{1}{236196} a^{10} + \frac{50389252959518968358719529848}{37139895994657683433617806879124513} a^{9} - \frac{2}{6561} a^{8} - \frac{11103642408515958536001872911307053}{5422424815220021781308199804352178898} a^{7} + \frac{23}{26244} a^{6} - \frac{1119467055663765411269727682098208}{100415274355926329283485181562077387} a^{5} + \frac{65}{8748} a^{4} - \frac{641689794374181668203860290507417}{14876336941618715449405212083270724} a^{3} - \frac{25}{324} a^{2} - \frac{30336764014747138188658499735771}{61219493586908294030474123799468} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{129077776}$, which has order $8260977664$ (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:  \( 5866539973.245223 \) (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

\(\Q(\sqrt{123}) \), \(\Q(\sqrt{-30}) \), \(\Q(\sqrt{-410}) \), \(\Q(\sqrt{-30}, \sqrt{123})\), 5.5.2825761.1, 10.10.81463101499118103552.1, 10.0.198690491461263667200000.1, 10.0.33523910081941606400000.1

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 R R R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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
2Data not computed
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
5Data not computed
41Data not computed