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Properties

Label 20.0.16389869939...5933.1
Degree $20$
Signature $[0, 10]$
Discriminant $7^{15}\cdot 11^{13}$
Root discriminant $20.45$
Ramified primes $7, 11$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_5\times C_5:D_4$ (as 20T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1024, -512, -2368, 2048, 5240, -1859, -7821, -1827, 6198, 4196, -1572, -2049, 330, 798, 0, -200, 6, 37, -3, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 - 3*x^18 + 37*x^17 + 6*x^16 - 200*x^15 + 798*x^13 + 330*x^12 - 2049*x^11 - 1572*x^10 + 4196*x^9 + 6198*x^8 - 1827*x^7 - 7821*x^6 - 1859*x^5 + 5240*x^4 + 2048*x^3 - 2368*x^2 - 512*x + 1024)
 
gp: K = bnfinit(x^20 - 4*x^19 - 3*x^18 + 37*x^17 + 6*x^16 - 200*x^15 + 798*x^13 + 330*x^12 - 2049*x^11 - 1572*x^10 + 4196*x^9 + 6198*x^8 - 1827*x^7 - 7821*x^6 - 1859*x^5 + 5240*x^4 + 2048*x^3 - 2368*x^2 - 512*x + 1024, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} - 3 x^{18} + 37 x^{17} + 6 x^{16} - 200 x^{15} + 798 x^{13} + 330 x^{12} - 2049 x^{11} - 1572 x^{10} + 4196 x^{9} + 6198 x^{8} - 1827 x^{7} - 7821 x^{6} - 1859 x^{5} + 5240 x^{4} + 2048 x^{3} - 2368 x^{2} - 512 x + 1024 \)

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:  \(163898699393368601103605933=7^{15}\cdot 11^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.45$
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:  $7, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{4} a^{16} + \frac{1}{4} a^{14} + \frac{1}{4} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{17} - \frac{3}{16} a^{15} - \frac{7}{16} a^{14} - \frac{3}{8} a^{13} - \frac{1}{8} a^{10} + \frac{1}{8} a^{9} + \frac{7}{16} a^{8} - \frac{1}{2} a^{7} + \frac{1}{4} a^{6} + \frac{3}{8} a^{5} + \frac{5}{16} a^{4} + \frac{7}{16} a^{3} - \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{64} a^{18} - \frac{3}{64} a^{16} + \frac{25}{64} a^{15} - \frac{11}{32} a^{14} - \frac{1}{2} a^{13} + \frac{15}{32} a^{11} + \frac{1}{32} a^{10} + \frac{7}{64} a^{9} - \frac{1}{8} a^{8} + \frac{1}{16} a^{7} + \frac{3}{32} a^{6} - \frac{11}{64} a^{5} + \frac{7}{64} a^{4} + \frac{25}{64} a^{3} + \frac{7}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{1207563658743716279284701506209024} a^{19} - \frac{90138933575371842598897167351}{150945457342964534910587688276128} a^{18} - \frac{28920721535347893446782638573459}{1207563658743716279284701506209024} a^{17} + \frac{75776686952642507906596031001345}{1207563658743716279284701506209024} a^{16} + \frac{50667207461988045227025283150225}{603781829371858139642350753104512} a^{15} - \frac{4986171008966702436420980313621}{37736364335741133727646922069032} a^{14} - \frac{3378029770428513003758640986651}{37736364335741133727646922069032} a^{13} + \frac{250560280619168918393097304014543}{603781829371858139642350753104512} a^{12} + \frac{265730913526857607991407749897113}{603781829371858139642350753104512} a^{11} + \frac{172309362236131977612597621758391}{1207563658743716279284701506209024} a^{10} + \frac{8244302920218396369906368096293}{75472728671482267455293844138064} a^{9} + \frac{116305799854595506463727370621029}{301890914685929069821175376552256} a^{8} - \frac{128353620285718189276436394620397}{603781829371858139642350753104512} a^{7} + \frac{181356788275856532063626725697381}{1207563658743716279284701506209024} a^{6} + \frac{333712617924911970403251597031375}{1207563658743716279284701506209024} a^{5} - \frac{285788149594392624285476738483327}{1207563658743716279284701506209024} a^{4} - \frac{120311022047877899116281104076435}{301890914685929069821175376552256} a^{3} - \frac{2588976521918518930840033186363}{9434091083935283431911730517258} a^{2} - \frac{1844682754859131319926347539882}{4717045541967641715955865258629} a - \frac{1256581358224001258352240463567}{4717045541967641715955865258629}$

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:  \( -1 \) (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:  \( 60077.16722605073 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_5\times C_5:D_4$ (as 20T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 200
The 65 conjugacy class representatives for $C_5\times C_5:D_4$ are not computed
Character table for $C_5\times C_5:D_4$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.3773.1, 10.0.246071287.1

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 ${\href{/LocalNumberField/2.10.0.1}{10} }{,}\,{\href{/LocalNumberField/2.5.0.1}{5} }^{2}$ $20$ $20$ R R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{10}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}$ $20$

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
7Data not computed
$11$11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.3$x^{5} + 33$$5$$1$$4$$C_5$$[\ ]_{5}$
11.10.5.2$x^{10} + 1331 x^{4} - 14641 x^{2} + 805255$$2$$5$$5$$C_{10}$$[\ ]_{2}^{5}$