↑

Properties

Label 20.0.46327297853...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{20}\cdot 5^{16}\cdot 6029^{7}$
Root discriminant $152.51$
Ramified primes $2, 5, 6029$
Class number $99357208$ (GRH)
Class group $[2, 2, 24839302]$ (GRH)
Galois group 20T796

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5478679059725, 0, 5130638907150, 0, 1897966067335, 0, 370949598950, 0, 42546340536, 0, 2982904259, 0, 128674960, 0, 3346810, 0, 49600, 0, 369, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 369*x^18 + 49600*x^16 + 3346810*x^14 + 128674960*x^12 + 2982904259*x^10 + 42546340536*x^8 + 370949598950*x^6 + 1897966067335*x^4 + 5130638907150*x^2 + 5478679059725)
 
gp: K = bnfinit(x^20 + 369*x^18 + 49600*x^16 + 3346810*x^14 + 128674960*x^12 + 2982904259*x^10 + 42546340536*x^8 + 370949598950*x^6 + 1897966067335*x^4 + 5130638907150*x^2 + 5478679059725, 1)
 

Normalized defining polynomial

\( x^{20} + 369 x^{18} + 49600 x^{16} + 3346810 x^{14} + 128674960 x^{12} + 2982904259 x^{10} + 42546340536 x^{8} + 370949598950 x^{6} + 1897966067335 x^{4} + 5130638907150 x^{2} + 5478679059725 \)

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:  \(46327297853378526155737329440000000000000000=2^{20}\cdot 5^{16}\cdot 6029^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $152.51$
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:  $2, 5, 6029$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{30145} a^{16} + \frac{369}{30145} a^{14} - \frac{2138}{6029} a^{12} + \frac{143}{6029} a^{10} - \frac{2809}{6029} a^{8} - \frac{3781}{30145} a^{6} - \frac{11014}{30145} a^{4}$, $\frac{1}{30145} a^{17} + \frac{369}{30145} a^{15} - \frac{2138}{6029} a^{13} + \frac{143}{6029} a^{11} - \frac{2809}{6029} a^{9} - \frac{3781}{30145} a^{7} - \frac{11014}{30145} a^{5}$, $\frac{1}{361929236338300145778587750375786855244944790383105} a^{18} - \frac{1249184819031957354048852610326334911574369739}{361929236338300145778587750375786855244944790383105} a^{16} - \frac{175533424901800599637028564735836155068005969600547}{361929236338300145778587750375786855244944790383105} a^{14} + \frac{18547102373998022882630603469255711211176115336243}{72385847267660029155717550075157371048988958076621} a^{12} - \frac{10292802077533659348282047644025300509868486797901}{72385847267660029155717550075157371048988958076621} a^{10} + \frac{86732268527868862850208809553380726327677847526284}{361929236338300145778587750375786855244944790383105} a^{8} - \frac{13672699658822616820264585952193511089055247911036}{361929236338300145778587750375786855244944790383105} a^{6} + \frac{18438853129059889995396166220699808849113270108}{60031387682584200659908401123865791216610514245} a^{4} + \frac{2575492865406995585058119100644188967707903340}{12006277536516840131981680224773158243322102849} a^{2} + \frac{777138723091040106828185096708017868815788}{1991421054323576070987175356572094583400581}$, $\frac{1}{361929236338300145778587750375786855244944790383105} a^{19} - \frac{1249184819031957354048852610326334911574369739}{361929236338300145778587750375786855244944790383105} a^{17} - \frac{175533424901800599637028564735836155068005969600547}{361929236338300145778587750375786855244944790383105} a^{15} + \frac{18547102373998022882630603469255711211176115336243}{72385847267660029155717550075157371048988958076621} a^{13} - \frac{10292802077533659348282047644025300509868486797901}{72385847267660029155717550075157371048988958076621} a^{11} + \frac{86732268527868862850208809553380726327677847526284}{361929236338300145778587750375786855244944790383105} a^{9} - \frac{13672699658822616820264585952193511089055247911036}{361929236338300145778587750375786855244944790383105} a^{7} + \frac{18438853129059889995396166220699808849113270108}{60031387682584200659908401123865791216610514245} a^{5} + \frac{2575492865406995585058119100644188967707903340}{12006277536516840131981680224773158243322102849} a^{3} + \frac{777138723091040106828185096708017868815788}{1991421054323576070987175356572094583400581} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{24839302}$, which has order $99357208$ (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:  \( 341439.528105 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T796:

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 t20n796 are not computed
Character table for t20n796 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 $20$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
$5$5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.6.5.1$x^{6} - 5$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
6029Data not computed