↑

Properties

Label 20.0.33630250781...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{16}\cdot 3^{16}\cdot 5^{23}$
Root discriminant $26.69$
Ramified primes $2, 3, 5$
Class number $5$ (GRH)
Class group $[5]$ (GRH)
Galois group $F_5$ (as 20T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1331, 10890, 37290, 65245, 56075, 9408, -21040, -9670, 8900, 6770, -2256, -2640, 275, 590, -30, -113, 30, -10, 10, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 5*x^19 + 10*x^18 - 10*x^17 + 30*x^16 - 113*x^15 - 30*x^14 + 590*x^13 + 275*x^12 - 2640*x^11 - 2256*x^10 + 6770*x^9 + 8900*x^8 - 9670*x^7 - 21040*x^6 + 9408*x^5 + 56075*x^4 + 65245*x^3 + 37290*x^2 + 10890*x + 1331)
 
gp: K = bnfinit(x^20 - 5*x^19 + 10*x^18 - 10*x^17 + 30*x^16 - 113*x^15 - 30*x^14 + 590*x^13 + 275*x^12 - 2640*x^11 - 2256*x^10 + 6770*x^9 + 8900*x^8 - 9670*x^7 - 21040*x^6 + 9408*x^5 + 56075*x^4 + 65245*x^3 + 37290*x^2 + 10890*x + 1331, 1)
 

Normalized defining polynomial

\( x^{20} - 5 x^{19} + 10 x^{18} - 10 x^{17} + 30 x^{16} - 113 x^{15} - 30 x^{14} + 590 x^{13} + 275 x^{12} - 2640 x^{11} - 2256 x^{10} + 6770 x^{9} + 8900 x^{8} - 9670 x^{7} - 21040 x^{6} + 9408 x^{5} + 56075 x^{4} + 65245 x^{3} + 37290 x^{2} + 10890 x + 1331 \)

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:  \(33630250781250000000000000000=2^{16}\cdot 3^{16}\cdot 5^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.69$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{11} a^{11} - \frac{1}{11} a$, $\frac{1}{11} a^{12} - \frac{1}{11} a^{2}$, $\frac{1}{11} a^{13} - \frac{1}{11} a^{3}$, $\frac{1}{11} a^{14} - \frac{1}{11} a^{4}$, $\frac{1}{11} a^{15} - \frac{1}{11} a^{5}$, $\frac{1}{121} a^{16} + \frac{1}{121} a^{15} + \frac{4}{121} a^{14} - \frac{1}{121} a^{13} - \frac{5}{121} a^{12} + \frac{4}{11} a^{10} - \frac{5}{11} a^{9} - \frac{3}{11} a^{8} - \frac{1}{11} a^{7} + \frac{54}{121} a^{6} - \frac{56}{121} a^{5} - \frac{37}{121} a^{4} + \frac{56}{121} a^{3} - \frac{28}{121} a^{2} - \frac{5}{11} a$, $\frac{1}{2299} a^{17} - \frac{3}{2299} a^{16} + \frac{7}{209} a^{15} - \frac{28}{2299} a^{14} - \frac{56}{2299} a^{13} + \frac{20}{2299} a^{12} - \frac{5}{209} a^{11} + \frac{1}{209} a^{10} - \frac{104}{209} a^{9} + \frac{5}{19} a^{8} - \frac{23}{2299} a^{7} - \frac{998}{2299} a^{6} - \frac{56}{209} a^{5} + \frac{1062}{2299} a^{4} + \frac{892}{2299} a^{3} + \frac{904}{2299} a^{2} - \frac{4}{209} a + \frac{3}{19}$, $\frac{1}{278179} a^{18} + \frac{21}{278179} a^{17} + \frac{24}{278179} a^{16} - \frac{3177}{278179} a^{15} - \frac{234}{278179} a^{14} - \frac{7613}{278179} a^{13} + \frac{1075}{25289} a^{12} + \frac{812}{25289} a^{11} + \frac{11700}{25289} a^{10} + \frac{5197}{25289} a^{9} + \frac{609}{14641} a^{8} + \frac{67211}{278179} a^{7} + \frac{91408}{278179} a^{6} - \frac{76441}{278179} a^{5} + \frac{68940}{278179} a^{4} + \frac{73327}{278179} a^{3} + \frac{10907}{25289} a^{2} + \frac{490}{2299} a - \frac{85}{209}$, $\frac{1}{7291006999153314415914529} a^{19} + \frac{11602474393209342816}{7291006999153314415914529} a^{18} + \frac{299594127511173893732}{7291006999153314415914529} a^{17} - \frac{12739160141633458273501}{7291006999153314415914529} a^{16} + \frac{159833510168861712979393}{7291006999153314415914529} a^{15} + \frac{2319547260565377986629}{383737210481753390311291} a^{14} - \frac{330875448439360579676172}{7291006999153314415914529} a^{13} + \frac{10438935400023625255061}{662818818104846765083139} a^{12} - \frac{394422491593424562703}{60256256191349705916649} a^{11} - \frac{215651320867309603704811}{662818818104846765083139} a^{10} - \frac{2402867441159870756452778}{7291006999153314415914529} a^{9} - \frac{46735313971413588285975}{383737210481753390311291} a^{8} + \frac{3106161305252212570765387}{7291006999153314415914529} a^{7} + \frac{1638912663279372220681130}{7291006999153314415914529} a^{6} - \frac{3564224845993864585718013}{7291006999153314415914529} a^{5} + \frac{701049212446925235407300}{7291006999153314415914529} a^{4} + \frac{1366484865816964233318515}{7291006999153314415914529} a^{3} + \frac{3544636622371161589265}{34885200952886671846481} a^{2} - \frac{17230405337845797286641}{60256256191349705916649} a - \frac{2541075667502514579014}{5477841471940882356059}$

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:  $9$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{34704080550713934194328}{7291006999153314415914529} a^{19} + \frac{195269511134846039557828}{7291006999153314415914529} a^{18} - \frac{462177931402987797129216}{7291006999153314415914529} a^{17} + \frac{592038570268379157985191}{7291006999153314415914529} a^{16} - \frac{1290941094233683499028924}{7291006999153314415914529} a^{15} + \frac{4537989881580364605381936}{7291006999153314415914529} a^{14} - \frac{1420426814166360511838952}{7291006999153314415914529} a^{13} - \frac{1886664169360231493473512}{662818818104846765083139} a^{12} + \frac{36590012081259043450620}{60256256191349705916649} a^{11} + \frac{8413097927406668412777456}{662818818104846765083139} a^{10} + \frac{17445599121664302393629516}{7291006999153314415914529} a^{9} - \frac{263404650816701732505510450}{7291006999153314415914529} a^{8} - \frac{137861802513877995273195864}{7291006999153314415914529} a^{7} + \frac{472317531920292241910703000}{7291006999153314415914529} a^{6} + \frac{434908252738021462584570492}{7291006999153314415914529} a^{5} - \frac{693683759052740603590492692}{7291006999153314415914529} a^{4} - \frac{1548798298783130483057131296}{7291006999153314415914529} a^{3} - \frac{103074569665775941855110996}{662818818104846765083139} a^{2} - \frac{2927433816484318380568308}{60256256191349705916649} a - \frac{26196369935495136614815}{5477841471940882356059} \) (order $10$)
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:  \( 3774223.93146 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.1.4050000.3 x5, 10.2.82012500000000.7 x5

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

Sibling fields

Degree 5 sibling: 5.1.4050000.3
Degree 10 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 R R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.1.0.1}{1} }^{20}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
2Data not computed
3Data not computed
5Data not computed