↑

Properties

Label 20.8.25490385156...0000.3
Degree $20$
Signature $[8, 6]$
Discriminant $2^{20}\cdot 5^{15}\cdot 6029^{5}$
Root discriminant $58.93$
Ramified primes $2, 5, 6029$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T797

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![753625, 0, 753625, 0, -406925, 0, -486825, 0, -39940, 0, 58920, 0, 16151, 0, 474, 0, -208, 0, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 9*x^18 - 208*x^16 + 474*x^14 + 16151*x^12 + 58920*x^10 - 39940*x^8 - 486825*x^6 - 406925*x^4 + 753625*x^2 + 753625)
 
gp: K = bnfinit(x^20 - 9*x^18 - 208*x^16 + 474*x^14 + 16151*x^12 + 58920*x^10 - 39940*x^8 - 486825*x^6 - 406925*x^4 + 753625*x^2 + 753625, 1)
 

Normalized defining polynomial

\( x^{20} - 9 x^{18} - 208 x^{16} + 474 x^{14} + 16151 x^{12} + 58920 x^{10} - 39940 x^{8} - 486825 x^{6} - 406925 x^{4} + 753625 x^{2} + 753625 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(254903851560926116768000000000000000=2^{20}\cdot 5^{15}\cdot 6029^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.93$
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:  $2, 5, 6029$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} + \frac{1}{5} a^{10} + \frac{2}{5} a^{8} - \frac{1}{5} a^{6} + \frac{1}{5} a^{4}$, $\frac{1}{5} a^{13} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} - \frac{1}{5} a^{7} + \frac{1}{5} a^{5}$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{10} + \frac{2}{5} a^{8} + \frac{2}{5} a^{6} - \frac{1}{5} a^{4}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{7} - \frac{1}{5} a^{5}$, $\frac{1}{25} a^{16} + \frac{1}{25} a^{14} + \frac{2}{25} a^{12} - \frac{6}{25} a^{10} - \frac{9}{25} a^{8} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4}$, $\frac{1}{25} a^{17} + \frac{1}{25} a^{15} + \frac{2}{25} a^{13} - \frac{6}{25} a^{11} - \frac{9}{25} a^{9} + \frac{1}{5} a^{7} + \frac{2}{5} a^{5}$, $\frac{1}{358705401717412473619025} a^{18} - \frac{884060597196061123268}{71741080343482494723805} a^{16} + \frac{20085042895943562963341}{358705401717412473619025} a^{14} - \frac{16432143110775001617458}{358705401717412473619025} a^{12} + \frac{129296122387140071931147}{358705401717412473619025} a^{10} - \frac{153889550994606436322381}{358705401717412473619025} a^{8} - \frac{34176016816336560981788}{71741080343482494723805} a^{6} + \frac{1476326711409697941151}{5518544641806345747985} a^{4} + \frac{69843001557648649419}{243190102859262693979} a^{2} - \frac{2150454579054149293334}{14348216068696498944761}$, $\frac{1}{1793527008587062368095125} a^{19} + \frac{9927913082716193328421}{1793527008587062368095125} a^{17} + \frac{34433258964640061908102}{1793527008587062368095125} a^{15} + \frac{12264289026617996272064}{1793527008587062368095125} a^{13} + \frac{43206825974961078262581}{1793527008587062368095125} a^{11} - \frac{128345779466057480088851}{358705401717412473619025} a^{9} - \frac{19827800747640062037027}{358705401717412473619025} a^{7} + \frac{1840457841987716397666}{5518544641806345747985} a^{5} + \frac{69843001557648649419}{1215950514296313469895} a^{3} - \frac{3299734129550129647619}{14348216068696498944761} a$

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:  $13$
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:  \( 6550559486.36 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T797:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 122880
The 108 conjugacy class representatives for t20n797 are not computed
Character table for t20n797 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.753625.1, 10.10.2839753203125.1

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

Sibling fields

Degree 20 siblings: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
6029Data not computed