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Properties

Label 20.0.26909016091...6769.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 31^{10}$
Root discriminant $83.46$
Ramified primes $3, 11, 31$
Class number $491784$ (GRH)
Class group $[491784]$ (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![8135688529, -3724011822, 6525498396, -2802216648, 2541271974, -996179610, 625432743, -219816938, 106768355, -33218012, 13139312, -3569442, 1176066, -273438, 75340, -14468, 3288, -484, 87, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 87*x^18 - 484*x^17 + 3288*x^16 - 14468*x^15 + 75340*x^14 - 273438*x^13 + 1176066*x^12 - 3569442*x^11 + 13139312*x^10 - 33218012*x^9 + 106768355*x^8 - 219816938*x^7 + 625432743*x^6 - 996179610*x^5 + 2541271974*x^4 - 2802216648*x^3 + 6525498396*x^2 - 3724011822*x + 8135688529)
 
gp: K = bnfinit(x^20 - 8*x^19 + 87*x^18 - 484*x^17 + 3288*x^16 - 14468*x^15 + 75340*x^14 - 273438*x^13 + 1176066*x^12 - 3569442*x^11 + 13139312*x^10 - 33218012*x^9 + 106768355*x^8 - 219816938*x^7 + 625432743*x^6 - 996179610*x^5 + 2541271974*x^4 - 2802216648*x^3 + 6525498396*x^2 - 3724011822*x + 8135688529, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 87 x^{18} - 484 x^{17} + 3288 x^{16} - 14468 x^{15} + 75340 x^{14} - 273438 x^{13} + 1176066 x^{12} - 3569442 x^{11} + 13139312 x^{10} - 33218012 x^{9} + 106768355 x^{8} - 219816938 x^{7} + 625432743 x^{6} - 996179610 x^{5} + 2541271974 x^{4} - 2802216648 x^{3} + 6525498396 x^{2} - 3724011822 x + 8135688529 \)

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:  \(269090160911006768826967986512025596769=3^{10}\cdot 11^{18}\cdot 31^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.46$
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:  $3, 11, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1023=3\cdot 11\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{1023}(1,·)$, $\chi_{1023}(776,·)$, $\chi_{1023}(652,·)$, $\chi_{1023}(526,·)$, $\chi_{1023}(464,·)$, $\chi_{1023}(466,·)$, $\chi_{1023}(497,·)$, $\chi_{1023}(404,·)$, $\chi_{1023}(590,·)$, $\chi_{1023}(280,·)$, $\chi_{1023}(991,·)$, $\chi_{1023}(32,·)$, $\chi_{1023}(743,·)$, $\chi_{1023}(619,·)$, $\chi_{1023}(557,·)$, $\chi_{1023}(559,·)$, $\chi_{1023}(433,·)$, $\chi_{1023}(371,·)$, $\chi_{1023}(247,·)$, $\chi_{1023}(1022,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{134} a^{18} + \frac{1}{67} a^{17} - \frac{27}{134} a^{16} - \frac{17}{134} a^{15} - \frac{31}{134} a^{14} + \frac{29}{134} a^{13} - \frac{13}{134} a^{12} - \frac{7}{134} a^{11} + \frac{6}{67} a^{10} + \frac{18}{67} a^{9} + \frac{17}{67} a^{8} - \frac{5}{67} a^{7} - \frac{32}{67} a^{6} + \frac{21}{67} a^{5} - \frac{45}{134} a^{4} + \frac{30}{67} a^{3} + \frac{27}{134} a^{2} + \frac{32}{67} a - \frac{1}{2}$, $\frac{1}{139313478166726225676718976573717578958139409002261840567525261078} a^{19} + \frac{189526887779638415099765858370347332631933342605588972008254633}{139313478166726225676718976573717578958139409002261840567525261078} a^{18} + \frac{12579414086640430122542677013225470821830411530381845097906673191}{69656739083363112838359488286858789479069704501130920283762630539} a^{17} - \frac{5771039761273676065574905945562698101579626405988289120752198173}{69656739083363112838359488286858789479069704501130920283762630539} a^{16} - \frac{6880200601381714381145064082985849325052621132482236116250231880}{69656739083363112838359488286858789479069704501130920283762630539} a^{15} - \frac{10687271283042794434648210871874171535654156030597923868702716655}{69656739083363112838359488286858789479069704501130920283762630539} a^{14} - \frac{15166055211865055704575811243360458934167798974861797210162684857}{139313478166726225676718976573717578958139409002261840567525261078} a^{13} + \frac{5661816732267083466843686974238718065909170191305702329817963264}{69656739083363112838359488286858789479069704501130920283762630539} a^{12} + \frac{9732571982935698053702642703888993699458288716113674328855505067}{139313478166726225676718976573717578958139409002261840567525261078} a^{11} - \frac{5501225765104129286641648159805320282818826575514133440860987754}{69656739083363112838359488286858789479069704501130920283762630539} a^{10} + \frac{9037369219349341069719099710403598639276964867713681497308124433}{139313478166726225676718976573717578958139409002261840567525261078} a^{9} + \frac{23445467225203930316734483764795846990118333194792246642375360206}{69656739083363112838359488286858789479069704501130920283762630539} a^{8} - \frac{8755964484400968801983953823433810392092520652577590554269752815}{69656739083363112838359488286858789479069704501130920283762630539} a^{7} - \frac{67884197131633243832758623901599982914264354831109753292557339091}{139313478166726225676718976573717578958139409002261840567525261078} a^{6} + \frac{25004537428319881023104099381095712777079550866213276374717534363}{139313478166726225676718976573717578958139409002261840567525261078} a^{5} + \frac{28877727528771446734808809614163997378142029572227834636577641514}{69656739083363112838359488286858789479069704501130920283762630539} a^{4} - \frac{21084418661343177403536732873067117449002920429932107480911391309}{69656739083363112838359488286858789479069704501130920283762630539} a^{3} + \frac{54244693346759594573579179984095030988299858346205609875685991435}{139313478166726225676718976573717578958139409002261840567525261078} a^{2} + \frac{33851058948755064371315808855062152576037337070294044184854777578}{69656739083363112838359488286858789479069704501130920283762630539} a + \frac{208634704216318573187232942424330909568923709641170945630805128}{1039652822139747952811335646072519245956264246285536123638248217}$

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

Class group and class number

$C_{491784}$, which has order $491784$ (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:  \( 125582.779517 \) (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

\(\Q(\sqrt{-1023}) \), \(\Q(\sqrt{-31}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-31}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.0.16403967840464902863.1, 10.0.6136912772340031.1, \(\Q(\zeta_{33})^+\)

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
3Data not computed
11Data not computed
$31$31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
31.10.5.2$x^{10} - 923521 x^{2} + 286291510$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$