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Properties

Label 20.0.93877031488...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{31}\cdot 17^{10}$
Root discriminant $49.96$
Ramified primes $5, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times F_5$ (as 20T9)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![168988496, 402067640, 498508960, 378693200, 184536915, 36486691, -18499285, -22103305, -9806600, -2451535, 240476, 323200, 158005, 24400, 4850, -919, 270, -35, 25, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496)
 
gp: K = bnfinit(x^20 - 5*x^19 + 25*x^18 - 35*x^17 + 270*x^16 - 919*x^15 + 4850*x^14 + 24400*x^13 + 158005*x^12 + 323200*x^11 + 240476*x^10 - 2451535*x^9 - 9806600*x^8 - 22103305*x^7 - 18499285*x^6 + 36486691*x^5 + 184536915*x^4 + 378693200*x^3 + 498508960*x^2 + 402067640*x + 168988496, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 25 x^{18} - 35 x^{17} + 270 x^{16} - 919 x^{15} + 4850 x^{14} + 24400 x^{13} + 158005 x^{12} + 323200 x^{11} + 240476 x^{10} - 2451535 x^{9} - 9806600 x^{8} - 22103305 x^{7} - 18499285 x^{6} + 36486691 x^{5} + 184536915 x^{4} + 378693200 x^{3} + 498508960 x^{2} + 402067640 x + 168988496 \)

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:  \(9387703148876316845417022705078125=5^{31}\cdot 17^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $49.96$
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:  $5, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{2} a^{6} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{76} a^{18} + \frac{7}{76} a^{16} - \frac{2}{19} a^{14} + \frac{5}{38} a^{13} - \frac{5}{38} a^{12} + \frac{1}{38} a^{11} + \frac{1}{76} a^{10} - \frac{1}{38} a^{9} - \frac{4}{19} a^{8} + \frac{5}{38} a^{7} - \frac{5}{19} a^{6} + \frac{2}{19} a^{5} - \frac{29}{76} a^{4} + \frac{9}{38} a^{3} - \frac{11}{76} a^{2} - \frac{9}{19} a + \frac{3}{19}$, $\frac{1}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{19} + \frac{8115113969455534951781988892180760684934801953018434045993632395187}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{18} + \frac{10801847081374795790237250702930492710858851355221992110742936730837}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{17} - \frac{213636318760062715601966761815787615425618587954274690961471165165583}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{16} + \frac{16555355102740202305739775448538663745929788562134832612804311763238}{230332238414849941032068795213804749870639465298829823185816133031221} a^{15} - \frac{139094398132643264585294628379724708778019298981084841019191278192151}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{14} - \frac{842628770417491290937425382635414004019299651059897833214501227729}{921328953659399764128275180855218999482557861195319292743264532124884} a^{13} - \frac{27403859127557514920850213379311214151516007359229845308062055436253}{460664476829699882064137590427609499741278930597659646371632266062442} a^{12} - \frac{374222255245502479311970971681218030938369435371893193690796874412505}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{11} + \frac{86898060059690868835614601680867263551261287989197097550330579576895}{460664476829699882064137590427609499741278930597659646371632266062442} a^{10} + \frac{95379842741702110535612727637434374929212576455999773051938745221307}{921328953659399764128275180855218999482557861195319292743264532124884} a^{9} - \frac{56349862803977788738152759856865306881542020713355863591649296803203}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{8} + \frac{61176008907793018586714395081294815298074785638115642751373478867173}{921328953659399764128275180855218999482557861195319292743264532124884} a^{7} - \frac{790226489237124404951396184854556210092946927552161025090699162080317}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{6} - \frac{146977703656593977666186096510158532961174982127552942757386907382993}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{5} + \frac{47885022183555814657706271368011376361969920673020760998448396223193}{96981995122042080434555282195286210471848195915296767657185740223672} a^{4} - \frac{507548804278599381085360496804623831946394441028109630702549690262723}{1842657907318799528256550361710437998965115722390638585486529064249768} a^{3} + \frac{101374436050883109287836543775782738537449925569809180067712362900268}{230332238414849941032068795213804749870639465298829823185816133031221} a^{2} - \frac{31576328811327137293174387292184960353903475236180166563393383693}{230332238414849941032068795213804749870639465298829823185816133031221} a + \frac{93819010693431326653789212025193937154883142497110761357023547670024}{230332238414849941032068795213804749870639465298829823185816133031221}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 265705502.8157561 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times F_5$ (as 20T9):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 40
The 10 conjugacy class representatives for $C_2\times F_5$
Character table for $C_2\times F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.36125.1, 5.1.78125.1, 10.2.30517578125.1

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

Sibling fields

Galois closure: data not computed
Degree 10 siblings: data not computed
Degree 20 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
5Data not computed
$17$17.4.2.2$x^{4} - 17 x^{2} + 867$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$