↑

Properties

Label 20.0.25000000000...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{30}$
Root discriminant $29.51$
Ramified primes $2, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_5\wr C_2$ (as 20T50)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![164, 180, 550, 1190, 1525, 2016, 1670, 650, 1105, 1050, 72, 80, 400, 130, 0, -8, 45, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 45*x^16 - 8*x^15 + 130*x^13 + 400*x^12 + 80*x^11 + 72*x^10 + 1050*x^9 + 1105*x^8 + 650*x^7 + 1670*x^6 + 2016*x^5 + 1525*x^4 + 1190*x^3 + 550*x^2 + 180*x + 164)
 
gp: K = bnfinit(x^20 + 45*x^16 - 8*x^15 + 130*x^13 + 400*x^12 + 80*x^11 + 72*x^10 + 1050*x^9 + 1105*x^8 + 650*x^7 + 1670*x^6 + 2016*x^5 + 1525*x^4 + 1190*x^3 + 550*x^2 + 180*x + 164, 1)
 

Normalized defining polynomial

\( x^{20} + 45 x^{16} - 8 x^{15} + 130 x^{13} + 400 x^{12} + 80 x^{11} + 72 x^{10} + 1050 x^{9} + 1105 x^{8} + 650 x^{7} + 1670 x^{6} + 2016 x^{5} + 1525 x^{4} + 1190 x^{3} + 550 x^{2} + 180 x + 164 \)

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:  \(250000000000000000000000000000=2^{28}\cdot 5^{30}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.51$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{10} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{10} a^{15} + \frac{1}{5} a^{5} - \frac{1}{2} a^{3} + \frac{1}{5}$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{6} - \frac{1}{2} a^{4} + \frac{1}{5} a$, $\frac{1}{50} a^{17} - \frac{1}{50} a^{16} - \frac{1}{50} a^{15} + \frac{2}{5} a^{12} - \frac{2}{5} a^{11} + \frac{1}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{25} a^{7} - \frac{6}{25} a^{6} + \frac{13}{50} a^{5} + \frac{1}{10} a^{4} - \frac{1}{2} a^{3} - \frac{4}{25} a^{2} + \frac{9}{25} a - \frac{6}{25}$, $\frac{1}{650} a^{18} - \frac{3}{325} a^{17} - \frac{1}{25} a^{16} - \frac{1}{130} a^{15} + \frac{1}{65} a^{14} - \frac{4}{13} a^{13} + \frac{32}{65} a^{12} + \frac{8}{65} a^{11} + \frac{23}{65} a^{10} - \frac{2}{65} a^{9} - \frac{79}{325} a^{8} + \frac{124}{325} a^{7} - \frac{77}{650} a^{6} - \frac{5}{13} a^{5} - \frac{9}{65} a^{4} - \frac{53}{650} a^{3} - \frac{2}{25} a^{2} + \frac{54}{325} a - \frac{3}{13}$, $\frac{1}{122019226527998976603561550} a^{19} - \frac{8141259350725293014471}{61009613263999488301780775} a^{18} - \frac{1194333776025329977653459}{122019226527998976603561550} a^{17} - \frac{268501322694274150068582}{12201922652799897660356155} a^{16} - \frac{278221424075879312327787}{9386094348307613584889350} a^{15} - \frac{606998176495185914970744}{12201922652799897660356155} a^{14} - \frac{5404388245083442383529652}{12201922652799897660356155} a^{13} + \frac{3998038548655677465780601}{12201922652799897660356155} a^{12} - \frac{195761606999183162835886}{938609434830761358488935} a^{11} + \frac{2097117193012890326979368}{12201922652799897660356155} a^{10} + \frac{622054512838562914654796}{61009613263999488301780775} a^{9} + \frac{23418845368005018799310383}{61009613263999488301780775} a^{8} + \frac{39483342418618454902726287}{122019226527998976603561550} a^{7} - \frac{515663314106126189717562}{12201922652799897660356155} a^{6} - \frac{20108044095189954644090407}{122019226527998976603561550} a^{5} - \frac{20575105024580657418719494}{61009613263999488301780775} a^{4} + \frac{60856864280581639829938741}{122019226527998976603561550} a^{3} + \frac{4174887686377536121394351}{61009613263999488301780775} a^{2} + \frac{1273229604604672009241067}{12201922652799897660356155} a - \frac{17654964047598186793502441}{61009613263999488301780775}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{5742750047529375}{15500754466467302894} a^{19} - \frac{9505881841100515}{15500754466467302894} a^{18} + \frac{33737828709924875}{7750377233233651447} a^{17} - \frac{64629805816972425}{15500754466467302894} a^{16} + \frac{308766807711967971}{15500754466467302894} a^{15} - \frac{262220081680023300}{7750377233233651447} a^{14} + \frac{1578579226413240045}{7750377233233651447} a^{13} - \frac{1363259712033778375}{7750377233233651447} a^{12} + \frac{1923591562251311595}{7750377233233651447} a^{11} + \frac{1369937745953526927}{7750377233233651447} a^{10} + \frac{10287160881186007450}{7750377233233651447} a^{9} - \frac{5001890647910414595}{7750377233233651447} a^{8} + \frac{9977517969900355075}{15500754466467302894} a^{7} + \frac{48940848069631618515}{15500754466467302894} a^{6} + \frac{10435355438486258857}{7750377233233651447} a^{5} + \frac{13908210640647421175}{15500754466467302894} a^{4} + \frac{65446448993574849015}{15500754466467302894} a^{3} + \frac{19197799794829582300}{7750377233233651447} a^{2} + \frac{11009259228910777610}{7750377233233651447} a + \frac{5613696519333453119}{7750377233233651447} \) (order $4$)
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:  \( 16526624.2831 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_5\wr C_2$ (as 20T50):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 14 conjugacy class representatives for $D_5\wr C_2$
Character table for $D_5\wr C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(i, \sqrt{5})\), 10.0.100000000000000.1

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

Sibling fields

Degree 10 siblings: data not computed
Degree 20 siblings: data not computed
Degree 25 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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{10}$

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
$2$2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
2.8.12.13$x^{8} + 12 x^{4} + 16$$4$$2$$12$$D_4$$[2, 2]^{2}$
$5$5.10.13.2$x^{10} + 10 x^{4} + 5$$10$$1$$13$$D_{10}$$[3/2]_{2}^{2}$
5.10.17.6$x^{10} - 20 x^{8} + 5$$10$$1$$17$$D_{10}$$[2]_{2}^{2}$