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Properties

Label 20.0.38319164111...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{30}\cdot 5^{34}\cdot 19^{10}$
Root discriminant $190.18$
Ramified primes $2, 5, 19$
Class number $24057220$ (GRH)
Class group $[22, 1093510]$ (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![9891422465, -6013728850, 7813521175, -3915754450, 2783495775, -1091599580, 539014250, -164582500, 60023285, -13885350, 4161641, -873790, 260725, -69700, 20610, -3212, -185, 150, 15, -10, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^19 + 15*x^18 + 150*x^17 - 185*x^16 - 3212*x^15 + 20610*x^14 - 69700*x^13 + 260725*x^12 - 873790*x^11 + 4161641*x^10 - 13885350*x^9 + 60023285*x^8 - 164582500*x^7 + 539014250*x^6 - 1091599580*x^5 + 2783495775*x^4 - 3915754450*x^3 + 7813521175*x^2 - 6013728850*x + 9891422465)
 
gp: K = bnfinit(x^20 - 10*x^19 + 15*x^18 + 150*x^17 - 185*x^16 - 3212*x^15 + 20610*x^14 - 69700*x^13 + 260725*x^12 - 873790*x^11 + 4161641*x^10 - 13885350*x^9 + 60023285*x^8 - 164582500*x^7 + 539014250*x^6 - 1091599580*x^5 + 2783495775*x^4 - 3915754450*x^3 + 7813521175*x^2 - 6013728850*x + 9891422465, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{19} + 15 x^{18} + 150 x^{17} - 185 x^{16} - 3212 x^{15} + 20610 x^{14} - 69700 x^{13} + 260725 x^{12} - 873790 x^{11} + 4161641 x^{10} - 13885350 x^{9} + 60023285 x^{8} - 164582500 x^{7} + 539014250 x^{6} - 1091599580 x^{5} + 2783495775 x^{4} - 3915754450 x^{3} + 7813521175 x^{2} - 6013728850 x + 9891422465 \)

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:  \(3831916411125625000000000000000000000000000000=2^{30}\cdot 5^{34}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $190.18$
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:  $2, 5, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3800=2^{3}\cdot 5^{2}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{3800}(1,·)$, $\chi_{3800}(3269,·)$, $\chi_{3800}(2241,·)$, $\chi_{3800}(1481,·)$, $\chi_{3800}(2509,·)$, $\chi_{3800}(721,·)$, $\chi_{3800}(1749,·)$, $\chi_{3800}(3001,·)$, $\chi_{3800}(989,·)$, $\chi_{3800}(229,·)$, $\chi_{3800}(3041,·)$, $\chi_{3800}(2469,·)$, $\chi_{3800}(1521,·)$, $\chi_{3800}(2281,·)$, $\chi_{3800}(1709,·)$, $\chi_{3800}(3229,·)$, $\chi_{3800}(3761,·)$, $\chi_{3800}(949,·)$, $\chi_{3800}(761,·)$, $\chi_{3800}(189,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{6} + \frac{1}{4} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{56} a^{12} - \frac{3}{28} a^{11} + \frac{3}{56} a^{10} - \frac{1}{28} a^{9} + \frac{1}{14} a^{8} - \frac{1}{4} a^{7} + \frac{3}{56} a^{6} - \frac{3}{28} a^{5} - \frac{13}{28} a^{3} + \frac{23}{56} a^{2} - \frac{11}{28} a - \frac{17}{56}$, $\frac{1}{56} a^{13} - \frac{5}{56} a^{11} + \frac{1}{28} a^{10} + \frac{3}{28} a^{9} - \frac{1}{14} a^{8} + \frac{3}{56} a^{7} - \frac{1}{28} a^{6} + \frac{3}{28} a^{5} - \frac{3}{14} a^{4} - \frac{3}{8} a^{3} + \frac{9}{28} a^{2} + \frac{5}{56} a + \frac{3}{7}$, $\frac{1}{56} a^{14} - \frac{1}{8} a^{10} - \frac{5}{56} a^{8} + \frac{3}{14} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{4} - \frac{5}{14} a^{2} + \frac{3}{14} a - \frac{1}{56}$, $\frac{1}{56} a^{15} - \frac{1}{8} a^{11} - \frac{5}{56} a^{9} - \frac{1}{28} a^{8} - \frac{1}{8} a^{7} + \frac{1}{8} a^{5} - \frac{1}{4} a^{4} - \frac{5}{14} a^{3} + \frac{3}{14} a^{2} - \frac{1}{56} a - \frac{1}{4}$, $\frac{1}{784} a^{16} + \frac{3}{392} a^{15} - \frac{1}{196} a^{12} + \frac{5}{392} a^{11} + \frac{23}{392} a^{10} - \frac{5}{392} a^{9} + \frac{5}{112} a^{8} + \frac{11}{56} a^{7} + \frac{15}{392} a^{6} + \frac{47}{392} a^{5} + \frac{81}{392} a^{4} - \frac{1}{196} a^{3} - \frac{15}{56} a^{2} - \frac{1}{392} a - \frac{9}{784}$, $\frac{1}{784} a^{17} + \frac{3}{392} a^{15} - \frac{1}{196} a^{13} + \frac{3}{392} a^{12} + \frac{1}{14} a^{11} + \frac{11}{392} a^{10} - \frac{59}{784} a^{9} - \frac{1}{14} a^{8} + \frac{23}{98} a^{7} - \frac{85}{392} a^{6} + \frac{15}{196} a^{5} - \frac{12}{49} a^{4} - \frac{51}{392} a^{3} + \frac{167}{392} a^{2} - \frac{207}{784} a + \frac{69}{392}$, $\frac{1}{819749866850002187974516200563984} a^{18} - \frac{9}{819749866850002187974516200563984} a^{17} - \frac{5127621888949261957327287615}{58553561917857299141036871468856} a^{16} + \frac{143573412890579334805164053271}{204937466712500546993629050140996} a^{15} - \frac{1007601063465984412779339157839}{409874933425001093987258100281992} a^{14} + \frac{289733999013087738753518891433}{58553561917857299141036871468856} a^{13} + \frac{1114843493377982700296850794845}{409874933425001093987258100281992} a^{12} - \frac{9994837148708080480540697520243}{204937466712500546993629050140996} a^{11} + \frac{1792706624867597841477659760203}{19063950391860515999407353501488} a^{10} + \frac{9283355298953540942424665340273}{819749866850002187974516200563984} a^{9} - \frac{48556359760463471202552824912563}{409874933425001093987258100281992} a^{8} - \frac{37390986857134533620524532952931}{204937466712500546993629050140996} a^{7} + \frac{763014521601368402948943969275}{102468733356250273496814525070498} a^{6} - \frac{101168985958979469310283525871875}{409874933425001093987258100281992} a^{5} - \frac{26169584258543190126328232807039}{204937466712500546993629050140996} a^{4} + \frac{13986966654140167771340411950287}{102468733356250273496814525070498} a^{3} - \frac{49686084047413691879759979254695}{117107123835714598282073742937712} a^{2} + \frac{322457122038007056996627064864563}{819749866850002187974516200563984} a - \frac{45947498714254437875638247826619}{102468733356250273496814525070498}$, $\frac{1}{77371352588007424660003346551134921690416} a^{19} + \frac{842715}{1381631296214418297500059759841695030186} a^{18} - \frac{20998495176357810451048064831131170587}{77371352588007424660003346551134921690416} a^{17} - \frac{5569113066042326030143150514563849337}{19342838147001856165000836637783730422604} a^{16} - \frac{85885380253948277569581554445308456415}{38685676294003712330001673275567460845208} a^{15} - \frac{541098101642218578554712108143288423}{19342838147001856165000836637783730422604} a^{14} + \frac{34754193169540540406425347701469809549}{9671419073500928082500418318891865211302} a^{13} + \frac{134891798669112400350083446894979316285}{19342838147001856165000836637783730422604} a^{12} - \frac{463477552037516823945302618328306160889}{11053050369715346380000478078733560241488} a^{11} + \frac{1421785298867508808516946254530990496183}{19342838147001856165000836637783730422604} a^{10} - \frac{6471220351337326528171824828988281859433}{77371352588007424660003346551134921690416} a^{9} + \frac{1580275352927939566707709682437311742005}{19342838147001856165000836637783730422604} a^{8} - \frac{7578333746554343425549234126624625329}{690815648107209148750029879920847515093} a^{7} - \frac{2431259617614702355067248797663192134569}{19342838147001856165000836637783730422604} a^{6} + \frac{52968801393826870312514885969991733733}{38685676294003712330001673275567460845208} a^{5} + \frac{1489059933656355174060777124094083531827}{9671419073500928082500418318891865211302} a^{4} + \frac{5321561538273450251165600030969827558225}{11053050369715346380000478078733560241488} a^{3} + \frac{1511046752421499958608947794185306643503}{9671419073500928082500418318891865211302} a^{2} - \frac{38423657226014187032075297246266172497865}{77371352588007424660003346551134921690416} a - \frac{3417753158819160348081704809778885404219}{9671419073500928082500418318891865211302}$

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

Class group and class number

$C_{22}\times C_{1093510}$, which has order $24057220$ (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:  \( 19344397.966990974 \) (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

\(\Q(\sqrt{10}) \), \(\Q(\sqrt{-19}) \), \(\Q(\sqrt{-190}) \), \(\Q(\sqrt{10}, \sqrt{-19})\), 5.5.390625.1, 10.10.25000000000000000.1, 10.0.377822723388671875.3, 10.0.61902475000000000000000.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
2Data not computed
$5$5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
5.10.17.29$x^{10} - 10 x^{8} + 35$$10$$1$$17$$C_{10}$$[2]_{2}$
19Data not computed