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Properties

Label 20.4.62703051848...8125.1
Degree $20$
Signature $[4, 8]$
Discriminant $5^{23}\cdot 47^{10}$
Root discriminant $43.64$
Ramified primes $5, 47$
Class number $8$ (GRH)
Class group $[2, 4]$ (GRH)
Galois group $C_2^4:C_5:C_4$ (as 20T88)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-22319, -15650, 2340, 22070, 15375, -26017, 13235, -10960, 5450, 660, -1091, 1915, -2825, 1775, -550, 27, 60, -5, -10, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 10*x^18 - 5*x^17 + 60*x^16 + 27*x^15 - 550*x^14 + 1775*x^13 - 2825*x^12 + 1915*x^11 - 1091*x^10 + 660*x^9 + 5450*x^8 - 10960*x^7 + 13235*x^6 - 26017*x^5 + 15375*x^4 + 22070*x^3 + 2340*x^2 - 15650*x - 22319)
 
gp: K = bnfinit(x^20 - 10*x^18 - 5*x^17 + 60*x^16 + 27*x^15 - 550*x^14 + 1775*x^13 - 2825*x^12 + 1915*x^11 - 1091*x^10 + 660*x^9 + 5450*x^8 - 10960*x^7 + 13235*x^6 - 26017*x^5 + 15375*x^4 + 22070*x^3 + 2340*x^2 - 15650*x - 22319, 1)
 

Normalized defining polynomial

\( x^{20} - 10 x^{18} - 5 x^{17} + 60 x^{16} + 27 x^{15} - 550 x^{14} + 1775 x^{13} - 2825 x^{12} + 1915 x^{11} - 1091 x^{10} + 660 x^{9} + 5450 x^{8} - 10960 x^{7} + 13235 x^{6} - 26017 x^{5} + 15375 x^{4} + 22070 x^{3} + 2340 x^{2} - 15650 x - 22319 \)

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:  \(627030518482089626789093017578125=5^{23}\cdot 47^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.64$
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:  $5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{94} a^{18} + \frac{4}{47} a^{17} - \frac{9}{94} a^{16} + \frac{15}{47} a^{15} + \frac{21}{94} a^{14} - \frac{3}{94} a^{13} - \frac{17}{94} a^{12} - \frac{5}{94} a^{11} + \frac{39}{94} a^{10} + \frac{2}{47} a^{9} - \frac{19}{47} a^{8} - \frac{37}{94} a^{7} + \frac{14}{47} a^{6} + \frac{4}{47} a^{5} + \frac{10}{47} a^{4} + \frac{3}{47} a^{3} + \frac{8}{47} a^{2} - \frac{35}{94} a - \frac{29}{94}$, $\frac{1}{3284744842349720740697482433574593549563046726} a^{19} - \frac{167348958655539816385405299415030652214731}{69888188135100441291435796459033905309852058} a^{18} - \frac{755712736901864853120089392798127021822166661}{3284744842349720740697482433574593549563046726} a^{17} - \frac{681224542431604568812133923307195869115562771}{3284744842349720740697482433574593549563046726} a^{16} + \frac{155171758739642092415455211816513299763311709}{3284744842349720740697482433574593549563046726} a^{15} - \frac{365192257933540085214070470211734067327832208}{1642372421174860370348741216787296774781523363} a^{14} + \frac{257572599650554524998524049553326019310503896}{1642372421174860370348741216787296774781523363} a^{13} - \frac{251188036392127000286085825018518224950076425}{1642372421174860370348741216787296774781523363} a^{12} - \frac{584919101334971734358143890188759440437507855}{1642372421174860370348741216787296774781523363} a^{11} - \frac{44166840566626182378436120062683870358247655}{3284744842349720740697482433574593549563046726} a^{10} + \frac{515883342366434270963312671283905037771151741}{1642372421174860370348741216787296774781523363} a^{9} + \frac{1431107911687317578749717546695280878813698125}{3284744842349720740697482433574593549563046726} a^{8} + \frac{312995298593810574678396016214221009313419667}{3284744842349720740697482433574593549563046726} a^{7} - \frac{342219159447133574380994133882017761268663578}{1642372421174860370348741216787296774781523363} a^{6} + \frac{367830732628826252024020789493988310731187364}{1642372421174860370348741216787296774781523363} a^{5} + \frac{739652043504582517512684441644007519028610266}{1642372421174860370348741216787296774781523363} a^{4} - \frac{180664097190135749232703503744429213562826297}{1642372421174860370348741216787296774781523363} a^{3} + \frac{1124519828101058270969000734543523265284269239}{3284744842349720740697482433574593549563046726} a^{2} - \frac{257090560036365255976821692249659479913066669}{1642372421174860370348741216787296774781523363} a - \frac{1108519000353005078885448066791238697163021363}{3284744842349720740697482433574593549563046726}$

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

Class group and class number

$C_{2}\times C_{4}$, which has order $8$ (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:  \( 36093748.9714 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^4:C_5:C_4$ (as 20T88):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 320
The 11 conjugacy class representatives for $C_2^4:C_5:C_4$
Character table for $C_2^4:C_5:C_4$

Intermediate fields

\(\Q(\sqrt{5}) \), 5.5.6903125.1, 10.10.238265673828125.1

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 16 sibling: data not computed
Degree 20 siblings: data not computed
Degree 32 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 ${\href{/LocalNumberField/2.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }$ ${\href{/LocalNumberField/3.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ R ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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
5Data not computed
$47$47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
47.4.2.2$x^{4} - 47 x^{2} + 28717$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$