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Properties

Label 20.0.22199191947...8125.2
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 31^{16}$
Root discriminant $52.16$
Ramified primes $5, 31$
Class number $400$ (GRH)
Class group $[2, 2, 10, 10]$ (GRH)
Galois group $C_{20}$ (as 20T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, -125, 2650, 445, 10616, -14164, 47983, -44226, 174168, -104714, 68890, -36325, 19301, -4980, 2045, -481, 138, -4, 13, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625)
 
gp: K = bnfinit(x^20 - x^19 + 13*x^18 - 4*x^17 + 138*x^16 - 481*x^15 + 2045*x^14 - 4980*x^13 + 19301*x^12 - 36325*x^11 + 68890*x^10 - 104714*x^9 + 174168*x^8 - 44226*x^7 + 47983*x^6 - 14164*x^5 + 10616*x^4 + 445*x^3 + 2650*x^2 - 125*x + 625, 1)
 

Normalized defining polynomial

\( x^{20} - x^{19} + 13 x^{18} - 4 x^{17} + 138 x^{16} - 481 x^{15} + 2045 x^{14} - 4980 x^{13} + 19301 x^{12} - 36325 x^{11} + 68890 x^{10} - 104714 x^{9} + 174168 x^{8} - 44226 x^{7} + 47983 x^{6} - 14164 x^{5} + 10616 x^{4} + 445 x^{3} + 2650 x^{2} - 125 x + 625 \)

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:  \(22199191947851112787734405517578125=5^{15}\cdot 31^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.16$
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:  $5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(155=5\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(2,·)$, $\chi_{155}(4,·)$, $\chi_{155}(97,·)$, $\chi_{155}(8,·)$, $\chi_{155}(128,·)$, $\chi_{155}(66,·)$, $\chi_{155}(78,·)$, $\chi_{155}(16,·)$, $\chi_{155}(132,·)$, $\chi_{155}(94,·)$, $\chi_{155}(32,·)$, $\chi_{155}(33,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(109,·)$, $\chi_{155}(47,·)$, $\chi_{155}(126,·)$, $\chi_{155}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5} a^{13} + \frac{2}{5} a^{12} + \frac{1}{5} a^{9} + \frac{1}{5} a^{7} + \frac{1}{5} a^{5} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{12} + \frac{1}{5} a^{10} - \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{3} - \frac{2}{5} a$, $\frac{1}{25} a^{15} + \frac{1}{25} a^{13} + \frac{1}{5} a^{12} + \frac{6}{25} a^{11} - \frac{12}{25} a^{10} + \frac{6}{25} a^{9} - \frac{7}{25} a^{8} + \frac{1}{25} a^{7} - \frac{2}{25} a^{6} - \frac{1}{5} a^{5} + \frac{3}{25} a^{4} - \frac{2}{5} a^{3} + \frac{3}{25} a^{2} + \frac{1}{5} a$, $\frac{1}{25} a^{16} + \frac{1}{25} a^{14} - \frac{4}{25} a^{12} - \frac{12}{25} a^{11} + \frac{6}{25} a^{10} - \frac{12}{25} a^{9} + \frac{1}{25} a^{8} - \frac{7}{25} a^{7} - \frac{1}{5} a^{6} - \frac{2}{25} a^{5} - \frac{2}{5} a^{4} + \frac{3}{25} a^{3} - \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{22506898179427525} a^{17} - \frac{294864270357121}{22506898179427525} a^{16} + \frac{10760498473873}{4501379635885505} a^{15} + \frac{909205899230709}{22506898179427525} a^{14} - \frac{210814021047053}{4501379635885505} a^{13} + \frac{10992802574110922}{22506898179427525} a^{12} + \frac{10047794312929867}{22506898179427525} a^{11} + \frac{4571424202498349}{22506898179427525} a^{10} - \frac{3247229745112748}{22506898179427525} a^{9} + \frac{504977962972054}{22506898179427525} a^{8} - \frac{1599328699750229}{22506898179427525} a^{7} + \frac{984027356852601}{4501379635885505} a^{6} - \frac{5510142347309073}{22506898179427525} a^{5} - \frac{563624833687709}{4501379635885505} a^{4} + \frac{11185173612652602}{22506898179427525} a^{3} + \frac{2942515553089197}{22506898179427525} a^{2} + \frac{2227239004620613}{4501379635885505} a + \frac{190931591142933}{900275927177101}$, $\frac{1}{22506898179427525} a^{18} + \frac{108192405962256}{22506898179427525} a^{16} + \frac{440880878229536}{22506898179427525} a^{15} + \frac{1757703920494636}{22506898179427525} a^{14} - \frac{1993203351602071}{22506898179427525} a^{13} - \frac{1670155948809464}{22506898179427525} a^{12} - \frac{9338181247683206}{22506898179427525} a^{11} + \frac{7652598014168409}{22506898179427525} a^{10} - \frac{10333008396483766}{22506898179427525} a^{9} + \frac{7408241507864408}{22506898179427525} a^{8} + \frac{4358306164450909}{22506898179427525} a^{7} + \frac{9240804825677303}{22506898179427525} a^{6} - \frac{5440226983366077}{22506898179427525} a^{5} + \frac{3634958918637348}{22506898179427525} a^{4} + \frac{429428454357341}{4501379635885505} a^{3} - \frac{1741011047262827}{22506898179427525} a^{2} - \frac{1436260478823882}{4501379635885505} a - \frac{68619080804835}{900275927177101}$, $\frac{1}{112534490897137625} a^{19} - \frac{1}{112534490897137625} a^{18} - \frac{2}{112534490897137625} a^{17} - \frac{533609215018744}{112534490897137625} a^{16} - \frac{129527128999212}{112534490897137625} a^{15} - \frac{10470790237586596}{112534490897137625} a^{14} + \frac{1342686089194689}{22506898179427525} a^{13} + \frac{7287767853137467}{22506898179427525} a^{12} - \frac{10295273957863169}{112534490897137625} a^{11} - \frac{10590228371976813}{22506898179427525} a^{10} + \frac{7906285212262434}{22506898179427525} a^{9} - \frac{11360769818760504}{112534490897137625} a^{8} - \frac{46137465089295612}{112534490897137625} a^{7} + \frac{37255316770823674}{112534490897137625} a^{6} + \frac{18526128933876088}{112534490897137625} a^{5} - \frac{46604583299806189}{112534490897137625} a^{4} + \frac{19873428633567271}{112534490897137625} a^{3} + \frac{9575852624661988}{22506898179427525} a^{2} - \frac{357946020788601}{4501379635885505} a - \frac{268705462053374}{900275927177101}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{10}\times C_{10}$, which has order $400$ (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:  \( \frac{34279214135504}{112534490897137625} a^{19} - \frac{33746819736904}{112534490897137625} a^{18} + \frac{438708656579752}{112534490897137625} a^{17} - \frac{134987278947616}{112534490897137625} a^{16} + \frac{4657061123692752}{112534490897137625} a^{15} - \frac{16532430736329149}{112534490897137625} a^{14} + \frac{13802449272393736}{22506898179427525} a^{13} - \frac{33611832457956384}{22506898179427525} a^{12} + \frac{651347367741984104}{112534490897137625} a^{11} - \frac{49034129077721512}{4501379635885505} a^{10} + \frac{454859818816663842}{22506898179427525} a^{9} - \frac{3533764481930165456}{112534490897137625} a^{8} + \frac{5877616099937095872}{112534490897137625} a^{7} - \frac{1492486849684316304}{112534490897137625} a^{6} + \frac{1619273651435864632}{112534490897137625} a^{5} - \frac{2003540773400225681}{112534490897137625} a^{4} + \frac{358256238326972864}{112534490897137625} a^{3} + \frac{3003466956584456}{22506898179427525} a^{2} + \frac{3577162892111824}{4501379635885505} a - \frac{33746819736904}{900275927177101} \) (order $10$)
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:  \( 24173706.8324 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{20}$ (as 20T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 20
The 20 conjugacy class representatives for $C_{20}$
Character table for $C_{20}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.923521.1, 10.10.2665284492003125.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $20$ $20$ R $20$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ $20$ $20$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ $20$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ $20$ $20$ $20$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$5$5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.2$x^{4} - 20$$4$$1$$3$$C_4$$[\ ]_{4}$
$31$31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$
31.5.4.1$x^{5} - 31$$5$$1$$4$$C_5$$[\ ]_{5}$