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Properties

Label 20.0.13600636709...5281.1
Degree $20$
Signature $[0, 10]$
Discriminant $3^{10}\cdot 11^{18}\cdot 23^{10}$
Root discriminant $71.89$
Ramified primes $3, 11, 23$
Class number $101376$ (GRH)
Class group $[4, 4, 4, 12, 132]$ (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![779300677, -383174430, 727021696, -347924432, 338081718, -151023234, 101680463, -41225306, 21607959, -7824820, 3365284, -1073170, 387206, -106790, 32388, -7492, 1876, -340, 67, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 8*x^19 + 67*x^18 - 340*x^17 + 1876*x^16 - 7492*x^15 + 32388*x^14 - 106790*x^13 + 387206*x^12 - 1073170*x^11 + 3365284*x^10 - 7824820*x^9 + 21607959*x^8 - 41225306*x^7 + 101680463*x^6 - 151023234*x^5 + 338081718*x^4 - 347924432*x^3 + 727021696*x^2 - 383174430*x + 779300677)
 
gp: K = bnfinit(x^20 - 8*x^19 + 67*x^18 - 340*x^17 + 1876*x^16 - 7492*x^15 + 32388*x^14 - 106790*x^13 + 387206*x^12 - 1073170*x^11 + 3365284*x^10 - 7824820*x^9 + 21607959*x^8 - 41225306*x^7 + 101680463*x^6 - 151023234*x^5 + 338081718*x^4 - 347924432*x^3 + 727021696*x^2 - 383174430*x + 779300677, 1)
 

Normalized defining polynomial

\( x^{20} - 8 x^{19} + 67 x^{18} - 340 x^{17} + 1876 x^{16} - 7492 x^{15} + 32388 x^{14} - 106790 x^{13} + 387206 x^{12} - 1073170 x^{11} + 3365284 x^{10} - 7824820 x^{9} + 21607959 x^{8} - 41225306 x^{7} + 101680463 x^{6} - 151023234 x^{5} + 338081718 x^{4} - 347924432 x^{3} + 727021696 x^{2} - 383174430 x + 779300677 \)

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:  \(13600636709996264858725830959545495281=3^{10}\cdot 11^{18}\cdot 23^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $71.89$
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:  $3, 11, 23$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(759=3\cdot 11\cdot 23\)
Dirichlet character group:    $\lbrace$$\chi_{759}(1,·)$, $\chi_{759}(643,·)$, $\chi_{759}(68,·)$, $\chi_{759}(70,·)$, $\chi_{759}(392,·)$, $\chi_{759}(461,·)$, $\chi_{759}(206,·)$, $\chi_{759}(530,·)$, $\chi_{759}(346,·)$, $\chi_{759}(91,·)$, $\chi_{759}(668,·)$, $\chi_{759}(413,·)$, $\chi_{759}(229,·)$, $\chi_{759}(553,·)$, $\chi_{759}(298,·)$, $\chi_{759}(367,·)$, $\chi_{759}(689,·)$, $\chi_{759}(691,·)$, $\chi_{759}(116,·)$, $\chi_{759}(758,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{79414079104701176390307522051083639006898174410469130886810266} a^{19} + \frac{9208723473775570628670508550913363543610186094108185131365064}{39707039552350588195153761025541819503449087205234565443405133} a^{18} + \frac{13337998338937537016660594781637256124269059019546517434517279}{79414079104701176390307522051083639006898174410469130886810266} a^{17} - \frac{6467834423321802878204041952186620319292179175244715843954815}{79414079104701176390307522051083639006898174410469130886810266} a^{16} - \frac{4069613752717530919885155171058548499977730863352976316356975}{39707039552350588195153761025541819503449087205234565443405133} a^{15} + \frac{15441168027906228851496239115133029546719822543646630623779169}{79414079104701176390307522051083639006898174410469130886810266} a^{14} + \frac{5968446359025964706860442674378348571195513600174146022292597}{79414079104701176390307522051083639006898174410469130886810266} a^{13} + \frac{8616125511045106090975751262930104620653996542584968504044613}{79414079104701176390307522051083639006898174410469130886810266} a^{12} + \frac{12204968412694980951459584221940459429841415342554832105259643}{79414079104701176390307522051083639006898174410469130886810266} a^{11} - \frac{6752234627315163815019271122638328514625793257778521941800478}{39707039552350588195153761025541819503449087205234565443405133} a^{10} + \frac{16561788277520409470132886910731547591274751107292712489203475}{79414079104701176390307522051083639006898174410469130886810266} a^{9} - \frac{12097989065711740474487053978075960988274838370702045698153001}{39707039552350588195153761025541819503449087205234565443405133} a^{8} + \frac{19507982272110224887286361786538213000354024561719114907687988}{39707039552350588195153761025541819503449087205234565443405133} a^{7} + \frac{51004501793724505735323844651950638392702871774331598136578}{201558576407871006066770360535745276667254249772764291590889} a^{6} - \frac{4037345725091540145639456414738492671103627408603154672301847}{79414079104701176390307522051083639006898174410469130886810266} a^{5} + \frac{23250609195582623946796617642960571391679772298091149102205341}{79414079104701176390307522051083639006898174410469130886810266} a^{4} + \frac{3987341530032241320486259458758910930391904183001048897679621}{39707039552350588195153761025541819503449087205234565443405133} a^{3} + \frac{9121274804184719654973978171491782286752282413378464812070657}{79414079104701176390307522051083639006898174410469130886810266} a^{2} - \frac{39442855509297972918716893483865538795657719901018240720457471}{79414079104701176390307522051083639006898174410469130886810266} a - \frac{55505862662662468183200237853424796591106819238102695935645}{201558576407871006066770360535745276667254249772764291590889}$

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

Class group and class number

$C_{4}\times C_{4}\times C_{4}\times C_{12}\times C_{132}$, which has order $101376$ (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

\(\Q(\sqrt{-759}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-23}) \), \(\Q(\sqrt{-23}, \sqrt{33})\), \(\Q(\zeta_{11})^+\), 10.0.3687904108026165159.1, \(\Q(\zeta_{33})^+\), 10.0.1379687283212183.1

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 ${\href{/LocalNumberField/2.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/5.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/47.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$3$3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
3.10.5.2$x^{10} - 81 x^{2} + 243$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$
11Data not computed
$23$23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$