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Properties

Label 20.4.39610783034...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{15}\cdot 7^{10}\cdot 11^{16}$
Root discriminant $60.24$
Ramified primes $5, 7, 11$
Class number $5$ (GRH)
Class group $[5]$ (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![18411920, -41740760, 5924040, 17902740, -18671249, -14910222, 873000, 3870405, 1143615, -443136, -270083, 29720, 34925, 385, -3918, -619, 390, 85, -25, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 3*x^19 - 25*x^18 + 85*x^17 + 390*x^16 - 619*x^15 - 3918*x^14 + 385*x^13 + 34925*x^12 + 29720*x^11 - 270083*x^10 - 443136*x^9 + 1143615*x^8 + 3870405*x^7 + 873000*x^6 - 14910222*x^5 - 18671249*x^4 + 17902740*x^3 + 5924040*x^2 - 41740760*x + 18411920)
 
gp: K = bnfinit(x^20 - 3*x^19 - 25*x^18 + 85*x^17 + 390*x^16 - 619*x^15 - 3918*x^14 + 385*x^13 + 34925*x^12 + 29720*x^11 - 270083*x^10 - 443136*x^9 + 1143615*x^8 + 3870405*x^7 + 873000*x^6 - 14910222*x^5 - 18671249*x^4 + 17902740*x^3 + 5924040*x^2 - 41740760*x + 18411920, 1)
 

Normalized defining polynomial

\( x^{20} - 3 x^{19} - 25 x^{18} + 85 x^{17} + 390 x^{16} - 619 x^{15} - 3918 x^{14} + 385 x^{13} + 34925 x^{12} + 29720 x^{11} - 270083 x^{10} - 443136 x^{9} + 1143615 x^{8} + 3870405 x^{7} + 873000 x^{6} - 14910222 x^{5} - 18671249 x^{4} + 17902740 x^{3} + 5924040 x^{2} - 41740760 x + 18411920 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(396107830343483954099825714111328125=5^{15}\cdot 7^{10}\cdot 11^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.24$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{9} + \frac{2}{5} a^{4}$, $\frac{1}{5} a^{10} + \frac{2}{5} a^{5}$, $\frac{1}{10} a^{11} - \frac{3}{10} a^{6} - \frac{1}{2} a$, $\frac{1}{50} a^{12} + \frac{1}{50} a^{11} + \frac{1}{25} a^{10} - \frac{1}{25} a^{9} - \frac{23}{50} a^{7} + \frac{7}{50} a^{6} - \frac{8}{25} a^{5} + \frac{8}{25} a^{4} - \frac{1}{5} a^{3} - \frac{3}{10} a^{2} - \frac{3}{10} a + \frac{2}{5}$, $\frac{1}{50} a^{13} + \frac{1}{50} a^{11} - \frac{2}{25} a^{10} + \frac{1}{25} a^{9} - \frac{3}{50} a^{8} - \frac{1}{5} a^{7} + \frac{7}{50} a^{6} + \frac{11}{25} a^{5} - \frac{3}{25} a^{4} - \frac{1}{10} a^{3} - \frac{3}{10} a - \frac{2}{5}$, $\frac{1}{50} a^{14} - \frac{1}{50} a^{9} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{11}{50} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{50} a^{15} - \frac{1}{50} a^{10} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{19}{50} a^{5} + \frac{2}{5} a^{2} - \frac{2}{5} a$, $\frac{1}{750} a^{16} - \frac{1}{125} a^{15} - \frac{1}{750} a^{14} - \frac{1}{150} a^{12} - \frac{7}{250} a^{11} - \frac{17}{375} a^{10} + \frac{7}{250} a^{9} + \frac{1}{25} a^{8} + \frac{3}{50} a^{7} - \frac{57}{250} a^{6} + \frac{36}{125} a^{5} - \frac{43}{250} a^{4} + \frac{12}{25} a^{3} + \frac{9}{50} a^{2} + \frac{1}{5} a - \frac{8}{75}$, $\frac{1}{750} a^{17} - \frac{7}{750} a^{15} - \frac{1}{125} a^{14} - \frac{1}{150} a^{13} - \frac{1}{125} a^{12} + \frac{7}{150} a^{11} + \frac{9}{250} a^{10} + \frac{11}{125} a^{9} - \frac{1}{10} a^{8} - \frac{6}{125} a^{7} - \frac{13}{50} a^{6} - \frac{11}{250} a^{5} + \frac{1}{125} a^{4} + \frac{23}{50} a^{3} + \frac{9}{50} a^{2} + \frac{59}{150} a - \frac{11}{25}$, $\frac{1}{1500} a^{18} - \frac{1}{1500} a^{17} - \frac{1}{1500} a^{16} - \frac{1}{300} a^{15} + \frac{1}{150} a^{14} - \frac{1}{1500} a^{13} - \frac{1}{375} a^{12} + \frac{1}{1500} a^{11} + \frac{1}{20} a^{10} - \frac{27}{500} a^{8} + \frac{1}{250} a^{7} - \frac{23}{500} a^{6} - \frac{27}{100} a^{5} + \frac{6}{25} a^{4} + \frac{59}{300} a^{2} + \frac{2}{15} a$, $\frac{1}{327402223219003196477169778141414133372013922241367000} a^{19} - \frac{15584080411694548978213803155564746338559288382451}{109134074406334398825723259380471377790671307413789000} a^{18} - \frac{47955363373972546031008523350277261058488542583057}{109134074406334398825723259380471377790671307413789000} a^{17} + \frac{60686203032277404437587616466034304780477131411043}{327402223219003196477169778141414133372013922241367000} a^{16} - \frac{110743799317781991638053470913852051388528875320011}{27283518601583599706430814845117844447667826853447250} a^{15} + \frac{12777372863247671935080679446778107492787865844881}{65480444643800639295433955628282826674402784448273400} a^{14} - \frac{170464582069342742066031807048338457511528247824301}{27283518601583599706430814845117844447667826853447250} a^{13} - \frac{1315514954015573136852210875753679552057900591114919}{327402223219003196477169778141414133372013922241367000} a^{12} - \frac{10797083064837634309303212050780380467933256427745313}{327402223219003196477169778141414133372013922241367000} a^{11} - \frac{1197387471183801661182259950548154382255545800895529}{163701111609501598238584889070707066686006961120683500} a^{10} + \frac{8491200304077981130655151578608513584363764459995971}{109134074406334398825723259380471377790671307413789000} a^{9} - \frac{2560428659339021638696928332098583913712328068616157}{54567037203167199412861629690235688895335653706894500} a^{8} - \frac{34002277412825135777047389998453894281280303427326943}{109134074406334398825723259380471377790671307413789000} a^{7} - \frac{25646571641892810431047740251833752316458200807839431}{109134074406334398825723259380471377790671307413789000} a^{6} - \frac{21857154018228101006648854501174903249437114160938083}{54567037203167199412861629690235688895335653706894500} a^{5} - \frac{6187122078087959327828572231682162182550787039820331}{54567037203167199412861629690235688895335653706894500} a^{4} + \frac{25926922226486490900836996688220649889090336384681863}{65480444643800639295433955628282826674402784448273400} a^{3} - \frac{102436410279427691187052280528941408592556693293157}{2182681488126687976514465187609427555813426148275780} a^{2} - \frac{1005673220298093478685195806124681108266121824079771}{8185055580475079911929244453535353334300348056034175} a - \frac{2329199463689710688841940814985488739786208843543348}{8185055580475079911929244453535353334300348056034175}$

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

Class group and class number

$C_{5}$, which has order $5$ (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:  $11$
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:  \( 3188661065.882411 \) (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

\(\Q(\sqrt{5}) \), 4.4.6125.1, 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 ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$