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Properties

Label 20.0.14674634925...3125.1
Degree $20$
Signature $[0, 10]$
Discriminant $5^{15}\cdot 37^{10}$
Root discriminant $20.34$
Ramified primes $5, 37$
Class number $2$
Class group $[2]$
Galois group $F_5$ (as 20T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![641, 686, 2191, 2746, 4572, 3726, 4941, 3946, 3207, 1496, 1176, 186, 242, 26, 76, -4, 27, -14, 11, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 4*x^19 + 11*x^18 - 14*x^17 + 27*x^16 - 4*x^15 + 76*x^14 + 26*x^13 + 242*x^12 + 186*x^11 + 1176*x^10 + 1496*x^9 + 3207*x^8 + 3946*x^7 + 4941*x^6 + 3726*x^5 + 4572*x^4 + 2746*x^3 + 2191*x^2 + 686*x + 641)
 
gp: K = bnfinit(x^20 - 4*x^19 + 11*x^18 - 14*x^17 + 27*x^16 - 4*x^15 + 76*x^14 + 26*x^13 + 242*x^12 + 186*x^11 + 1176*x^10 + 1496*x^9 + 3207*x^8 + 3946*x^7 + 4941*x^6 + 3726*x^5 + 4572*x^4 + 2746*x^3 + 2191*x^2 + 686*x + 641, 1)
 

Normalized defining polynomial

\( x^{20} - 4 x^{19} + 11 x^{18} - 14 x^{17} + 27 x^{16} - 4 x^{15} + 76 x^{14} + 26 x^{13} + 242 x^{12} + 186 x^{11} + 1176 x^{10} + 1496 x^{9} + 3207 x^{8} + 3946 x^{7} + 4941 x^{6} + 3726 x^{5} + 4572 x^{4} + 2746 x^{3} + 2191 x^{2} + 686 x + 641 \)

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:  \(146746349255915802001953125=5^{15}\cdot 37^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.34$
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:  $5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} + \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{4} a^{15} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{240} a^{16} - \frac{1}{8} a^{15} + \frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{15} a^{12} - \frac{1}{16} a^{11} + \frac{5}{24} a^{10} - \frac{3}{16} a^{9} + \frac{41}{240} a^{8} + \frac{1}{16} a^{7} - \frac{1}{48} a^{6} + \frac{7}{24} a^{5} - \frac{29}{240} a^{4} - \frac{1}{24} a^{3} + \frac{1}{8} a^{2} - \frac{5}{48} a - \frac{29}{240}$, $\frac{1}{240} a^{17} + \frac{1}{16} a^{15} + \frac{1}{16} a^{14} - \frac{7}{120} a^{13} - \frac{1}{16} a^{12} + \frac{1}{12} a^{11} + \frac{1}{16} a^{10} - \frac{49}{240} a^{9} + \frac{3}{16} a^{8} - \frac{7}{48} a^{7} - \frac{1}{12} a^{6} + \frac{31}{240} a^{5} - \frac{1}{6} a^{4} + \frac{1}{8} a^{3} - \frac{5}{48} a^{2} + \frac{61}{240} a - \frac{3}{8}$, $\frac{1}{37200} a^{18} - \frac{17}{37200} a^{17} + \frac{13}{18600} a^{16} + \frac{21}{248} a^{15} + \frac{256}{2325} a^{14} - \frac{1}{18600} a^{13} - \frac{4349}{37200} a^{12} - \frac{23}{744} a^{11} + \frac{981}{6200} a^{10} - \frac{9157}{37200} a^{9} - \frac{5749}{37200} a^{8} + \frac{13}{186} a^{7} + \frac{319}{9300} a^{6} - \frac{6517}{37200} a^{5} + \frac{12031}{37200} a^{4} - \frac{111}{496} a^{3} + \frac{107}{775} a^{2} - \frac{3011}{18600} a - \frac{14989}{37200}$, $\frac{1}{167665766085357659454000} a^{19} - \frac{706799386761315043}{167665766085357659454000} a^{18} + \frac{40228108494854159281}{55888588695119219818000} a^{17} - \frac{143695492746124894751}{167665766085357659454000} a^{16} - \frac{522021554896112207159}{5408573099527666434000} a^{15} + \frac{244355358888549307022}{10479110380334853715875} a^{14} + \frac{1645169019413775313591}{20958220760669707431750} a^{13} - \frac{2958739112330050581901}{167665766085357659454000} a^{12} + \frac{33183195543252153762761}{167665766085357659454000} a^{11} - \frac{5749904719079623719603}{27944294347559609909000} a^{10} + \frac{12327675301056420302983}{167665766085357659454000} a^{9} - \frac{7842035964209956317319}{41916441521339414863500} a^{8} - \frac{820466785910682230188}{3493036793444951238625} a^{7} + \frac{8011058200460170100033}{41916441521339414863500} a^{6} - \frac{2597843682874595580763}{41916441521339414863500} a^{5} + \frac{10054171066180699818399}{27944294347559609909000} a^{4} - \frac{26522782651085397523507}{83832883042678829727000} a^{3} + \frac{39198320662780616925917}{167665766085357659454000} a^{2} + \frac{46254910164415329388883}{167665766085357659454000} a - \frac{28340611811854936036511}{167665766085357659454000}$

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

Class group and class number

$C_{2}$, which has order $2$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 72096.60689 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_5$ (as 20T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 20
The 5 conjugacy class representatives for $F_5$
Character table for $F_5$

Intermediate fields

\(\Q(\sqrt{5}) \), 4.0.171125.1, 5.1.171125.1 x5, 10.2.146418828125.1 x5

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

Sibling fields

Degree 5 sibling: 5.1.171125.1
Degree 10 sibling: 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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{10}$ R ${\href{/LocalNumberField/41.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}$

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$5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
5.4.3.1$x^{4} - 5$$4$$1$$3$$C_4$$[\ ]_{4}$
$37$37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
37.4.2.2$x^{4} - 37 x^{2} + 6845$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$