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Properties

Label 20.0.41418647041...4992.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 17^{15}\cdot 53^{14}$
Root discriminant $269.69$
Ramified primes $2, 17, 53$
Class number $6407424$ (GRH)
Class group $[2, 2, 6, 12, 22248]$ (GRH)
Galois group $C_4:F_5$ (as 20T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15554125329, -24323038390, 14977761454, -4594563420, 6843858589, 245640690, 1787066994, 864149074, 321304814, 75736410, 38130743, 7067238, 2612317, 596526, 168540, 23822, 8449, -4, 177, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 177*x^18 - 4*x^17 + 8449*x^16 + 23822*x^15 + 168540*x^14 + 596526*x^13 + 2612317*x^12 + 7067238*x^11 + 38130743*x^10 + 75736410*x^9 + 321304814*x^8 + 864149074*x^7 + 1787066994*x^6 + 245640690*x^5 + 6843858589*x^4 - 4594563420*x^3 + 14977761454*x^2 - 24323038390*x + 15554125329)
 
gp: K = bnfinit(x^20 - 4*x^19 + 177*x^18 - 4*x^17 + 8449*x^16 + 23822*x^15 + 168540*x^14 + 596526*x^13 + 2612317*x^12 + 7067238*x^11 + 38130743*x^10 + 75736410*x^9 + 321304814*x^8 + 864149074*x^7 + 1787066994*x^6 + 245640690*x^5 + 6843858589*x^4 - 4594563420*x^3 + 14977761454*x^2 - 24323038390*x + 15554125329, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 177 x^{18} - 4 x^{17} + 8449 x^{16} + 23822 x^{15} + 168540 x^{14} + 596526 x^{13} + 2612317 x^{12} + 7067238 x^{11} + 38130743 x^{10} + 75736410 x^{9} + 321304814 x^{8} + 864149074 x^{7} + 1787066994 x^{6} + 245640690 x^{5} + 6843858589 x^{4} - 4594563420 x^{3} + 14977761454 x^{2} - 24323038390 x + 15554125329 \)

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:  \(4141864704128763454804479176602577540516903124992=2^{20}\cdot 17^{15}\cdot 53^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $269.69$
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:  $2, 17, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{14} + \frac{1}{3} a^{13} - \frac{1}{3} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{15} - \frac{4}{9} a^{14} - \frac{2}{9} a^{13} - \frac{4}{9} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{2}{9} a + \frac{1}{3}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{15} - \frac{4}{9} a^{14} + \frac{4}{9} a^{13} + \frac{4}{9} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{5} - \frac{2}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{37449} a^{18} - \frac{1754}{37449} a^{17} - \frac{227}{37449} a^{16} + \frac{183}{1387} a^{15} - \frac{2605}{12483} a^{14} - \frac{9034}{37449} a^{13} - \frac{15317}{37449} a^{12} - \frac{5735}{12483} a^{11} + \frac{3104}{12483} a^{10} - \frac{4366}{12483} a^{9} - \frac{16831}{37449} a^{8} + \frac{8527}{37449} a^{7} - \frac{17149}{37449} a^{6} - \frac{18146}{37449} a^{5} + \frac{436}{4161} a^{4} + \frac{3329}{37449} a^{3} - \frac{850}{37449} a^{2} + \frac{3335}{37449} a + \frac{3926}{12483}$, $\frac{1}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{19} - \frac{1123333170804544833482212575631952545468272879171749703819410059622056036541030330417172038345245}{121180316936633955782664960442150898834944615451528665522825891900363176927558020965196151054181003311} a^{18} + \frac{5501742588636944659811034539165810577417664166261225534194888914682536623421101427707833174987568230}{121180316936633955782664960442150898834944615451528665522825891900363176927558020965196151054181003311} a^{17} + \frac{7407389245933444887775948319623477490980723231926930734173654019512375191884252368289299075621161245}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{16} + \frac{3626169675745184934672633074241835229606008242340331709826467194681291983064556988217162584169124836}{121180316936633955782664960442150898834944615451528665522825891900363176927558020965196151054181003311} a^{15} - \frac{68769982733371714959415716265264927518764488362704128722490038701971666410492643816975037157269278390}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{14} - \frac{65032638970360344836615976509834516277191052941728514299115181289201155932124061048185379121244826462}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{13} + \frac{132342626737309884653066316124782448726106684672013763346677147199946789975934870883524843782209113199}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{12} - \frac{13164959911843334311004050982331895770607858395780986392719339817126781559408483838793456833840635617}{121180316936633955782664960442150898834944615451528665522825891900363176927558020965196151054181003311} a^{11} - \frac{1132695641885411734181511191906665296631037144105139167402607833779078303154346236616447963557981840}{40393438978877985260888320147383632944981538483842888507608630633454392309186006988398717018060334437} a^{10} - \frac{140821152003314309762588642309774193860239023256331700013568027388234606087625532621120924398787421973}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{9} - \frac{141219524553819770831490922162077092630464590594160373223155376050481167043525401423047443822447618931}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{8} - \frac{49458871701426420506333557915801716890689117596989464737253985731364754997483289255577129594703423481}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{7} - \frac{179747800169107664917485371851690248225489941532312327405036082157997686550448664092896947645019928039}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{6} + \frac{8164337738180876007410024700753507156536608934596756687547668694914991657689502988794050046666743010}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{5} - \frac{129316138305058285141407415738387723501864639405320449256690287696748307388374781939954151222611875936}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a^{4} + \frac{13769202165355240372285633199271551690011289112704474813768348009102980242335221712136208048148028214}{40393438978877985260888320147383632944981538483842888507608630633454392309186006988398717018060334437} a^{3} - \frac{29143727764447957141209559724089335185000515356227952333416589732937493992111400716591218370247770685}{121180316936633955782664960442150898834944615451528665522825891900363176927558020965196151054181003311} a^{2} - \frac{136298545995675967912223377988915434089102638697248184394806576787144530269748586508999970736817922896}{363540950809901867347994881326452696504833846354585996568477675701089530782674062895588453162543009933} a + \frac{49727935365082513014344236521778503633711011016395086876250334244914219045438833344062924485113329325}{121180316936633955782664960442150898834944615451528665522825891900363176927558020965196151054181003311}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{6}\times C_{12}\times C_{22248}$, which has order $6407424$ (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:  \( 8009866748.83 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4:F_5$ (as 20T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 14 conjugacy class representatives for $C_4:F_5$
Character table for $C_4:F_5$

Intermediate fields

\(\Q(\sqrt{901}) \), 4.0.220809872.1, 5.5.2382032.1, 10.10.426987989728139087104.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
2.8.8.1$x^{8} + 28 x^{4} + 144$$2$$4$$8$$C_4\times C_2$$[2]^{4}$
$17$17.4.3.2$x^{4} - 153$$4$$1$$3$$C_4$$[\ ]_{4}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
17.8.6.1$x^{8} - 119 x^{4} + 23409$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53Data not computed