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Properties

Label 20.0.47929047471...3125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 7^{10}\cdot 11^{18}$
Root discriminant $76.56$
Ramified primes $5, 7, 11$
Class number $80$ (GRH)
Class group $[2, 2, 20]$ (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![778286125, -767760000, 1251134700, -1118784220, 871193851, -515816388, 263443921, -108630102, 41055054, -13640902, 4600432, -1471866, 424696, -96020, 14657, -846, -231, 30, 10, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 6*x^19 + 10*x^18 + 30*x^17 - 231*x^16 - 846*x^15 + 14657*x^14 - 96020*x^13 + 424696*x^12 - 1471866*x^11 + 4600432*x^10 - 13640902*x^9 + 41055054*x^8 - 108630102*x^7 + 263443921*x^6 - 515816388*x^5 + 871193851*x^4 - 1118784220*x^3 + 1251134700*x^2 - 767760000*x + 778286125)
 
gp: K = bnfinit(x^20 - 6*x^19 + 10*x^18 + 30*x^17 - 231*x^16 - 846*x^15 + 14657*x^14 - 96020*x^13 + 424696*x^12 - 1471866*x^11 + 4600432*x^10 - 13640902*x^9 + 41055054*x^8 - 108630102*x^7 + 263443921*x^6 - 515816388*x^5 + 871193851*x^4 - 1118784220*x^3 + 1251134700*x^2 - 767760000*x + 778286125, 1)
 

Normalized defining polynomial

\( x^{20} - 6 x^{19} + 10 x^{18} + 30 x^{17} - 231 x^{16} - 846 x^{15} + 14657 x^{14} - 96020 x^{13} + 424696 x^{12} - 1471866 x^{11} + 4600432 x^{10} - 13640902 x^{9} + 41055054 x^{8} - 108630102 x^{7} + 263443921 x^{6} - 515816388 x^{5} + 871193851 x^{4} - 1118784220 x^{3} + 1251134700 x^{2} - 767760000 x + 778286125 \)

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:  \(47929047471561558446078911407470703125=5^{15}\cdot 7^{10}\cdot 11^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $76.56$
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:  $5, 7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{8} a^{15} - \frac{1}{8} a^{14} - \frac{1}{8} a^{13} + \frac{1}{8} a^{12} + \frac{1}{8} a^{11} + \frac{1}{8} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} + \frac{3}{8} a^{5} - \frac{1}{8} a^{4} - \frac{3}{8} a^{3} - \frac{3}{8} a^{2} - \frac{1}{8} a - \frac{1}{8}$, $\frac{1}{48} a^{16} - \frac{1}{24} a^{15} + \frac{1}{12} a^{14} - \frac{5}{24} a^{13} - \frac{1}{6} a^{12} + \frac{1}{6} a^{11} + \frac{5}{48} a^{10} + \frac{1}{24} a^{9} - \frac{1}{8} a^{8} - \frac{1}{24} a^{7} - \frac{1}{48} a^{6} - \frac{1}{3} a^{5} + \frac{1}{24} a^{4} + \frac{1}{6} a^{3} - \frac{1}{24} a^{2} + \frac{5}{12} a - \frac{19}{48}$, $\frac{1}{96} a^{17} - \frac{1}{96} a^{16} + \frac{1}{48} a^{15} - \frac{1}{16} a^{14} + \frac{1}{16} a^{13} - \frac{11}{96} a^{11} - \frac{17}{96} a^{10} + \frac{5}{24} a^{9} - \frac{1}{12} a^{8} - \frac{1}{32} a^{7} + \frac{31}{96} a^{6} - \frac{19}{48} a^{5} + \frac{17}{48} a^{4} + \frac{5}{16} a^{3} - \frac{5}{16} a^{2} + \frac{25}{96} a + \frac{29}{96}$, $\frac{1}{960} a^{18} + \frac{1}{240} a^{17} + \frac{1}{192} a^{16} - \frac{1}{16} a^{15} + \frac{13}{120} a^{14} + \frac{47}{480} a^{13} + \frac{39}{320} a^{12} + \frac{1}{24} a^{11} - \frac{169}{960} a^{10} + \frac{17}{80} a^{9} + \frac{197}{960} a^{8} + \frac{1}{20} a^{7} - \frac{371}{960} a^{6} - \frac{23}{240} a^{5} + \frac{9}{40} a^{4} - \frac{31}{120} a^{3} + \frac{97}{320} a^{2} - \frac{31}{96} a + \frac{37}{192}$, $\frac{1}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{19} - \frac{207816152789054403405833519344495970678196518745251160393533528546143756557}{641834390071454118280461584183808342660052390494665846551003179460151970876800} a^{18} + \frac{125651249728461608371175612237409051749294567416669023385607202105038626265}{77020126808574494193655390102057001119206286859359901586120381535218236505216} a^{17} - \frac{2112584989214597780887496798877714279772512033335838361447703911178737721839}{385100634042872470968276950510285005596031434296799507930601907676091182526080} a^{16} - \frac{23997132178679269456167682681645260606826467429152055277470559577350118026189}{481375792553590588710346188137856256995039292870999384913252384595113978157600} a^{15} + \frac{15727608386777237519455339856658490568579165578451528160171757697257678164649}{320917195035727059140230792091904171330026195247332923275501589730075985438400} a^{14} + \frac{44027561159120795894084165973894609163700956726529598107046003272578188701947}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{13} - \frac{15392133026493520343842039861521111466144634194816403888286181888440573862807}{77020126808574494193655390102057001119206286859359901586120381535218236505216} a^{12} - \frac{392156430686859126506893826325847448937654593726012936848872744642261373039529}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{11} - \frac{162966559868253272732007013930191058720516347653796835215790399042586225797481}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{10} - \frac{136754373330751103665131631960828149531446130342776602608485128173873301302101}{641834390071454118280461584183808342660052390494665846551003179460151970876800} a^{9} + \frac{40014591996950078157821908542220005372682807192668186715368925198826493755993}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{8} + \frac{38525841611039505975214119618422903553320266477476639116723011833421056605909}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{7} - \frac{191658131109346084585346749678854955040729696041197191241113562237964412422529}{641834390071454118280461584183808342660052390494665846551003179460151970876800} a^{6} + \frac{124945823009491952477257747232885001061277709263417231316667628331932714706719}{481375792553590588710346188137856256995039292870999384913252384595113978157600} a^{5} + \frac{1919908084077492060538433024274964405415970090956181556650464734739930662867}{120343948138397647177586547034464064248759823217749846228313096148778494539400} a^{4} - \frac{299969676845407752360238967907866642917497009935939019891472079221413312669029}{1925503170214362354841384752551425027980157171483997539653009538380455912630400} a^{3} - \frac{60736952325496420079555889378377530924249415647462443968639971823761535072969}{128366878014290823656092316836761668532010478098933169310200635892030394175360} a^{2} - \frac{1773768227563869605369135545058681199112954782315378553684650824685526894359}{25673375602858164731218463367352333706402095619786633862040127178406078835072} a + \frac{27480880976738379775807046605467432528749367658366821971769532365229381063949}{77020126808574494193655390102057001119206286859359901586120381535218236505216}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{20}$, which has order $80$ (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:  \( 818536127.5632833 \) (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

\(\Q(\sqrt{5}) \), 4.0.741125.2, 5.1.1830125.1, 10.2.16746787578125.1

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 R R ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$5$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
$7$7.4.2.2$x^{4} - 7 x^{2} + 147$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
$11$11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$
11.10.9.8$x^{10} + 33$$10$$1$$9$$C_{10}$$[\ ]_{10}$