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Properties

Label 20.0.97531030166...0000.3
Degree $20$
Signature $[0, 10]$
Discriminant $2^{24}\cdot 5^{12}\cdot 47^{8}$
Root discriminant $28.15$
Ramified primes $2, 5, 47$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -8, 74, -423, 1756, -4910, 9115, -11913, 12227, -10813, 8704, -6412, 4197, -2427, 1280, -615, 266, -97, 29, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 7*x^19 + 29*x^18 - 97*x^17 + 266*x^16 - 615*x^15 + 1280*x^14 - 2427*x^13 + 4197*x^12 - 6412*x^11 + 8704*x^10 - 10813*x^9 + 12227*x^8 - 11913*x^7 + 9115*x^6 - 4910*x^5 + 1756*x^4 - 423*x^3 + 74*x^2 - 8*x + 1)
 
gp: K = bnfinit(x^20 - 7*x^19 + 29*x^18 - 97*x^17 + 266*x^16 - 615*x^15 + 1280*x^14 - 2427*x^13 + 4197*x^12 - 6412*x^11 + 8704*x^10 - 10813*x^9 + 12227*x^8 - 11913*x^7 + 9115*x^6 - 4910*x^5 + 1756*x^4 - 423*x^3 + 74*x^2 - 8*x + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 7 x^{19} + 29 x^{18} - 97 x^{17} + 266 x^{16} - 615 x^{15} + 1280 x^{14} - 2427 x^{13} + 4197 x^{12} - 6412 x^{11} + 8704 x^{10} - 10813 x^{9} + 12227 x^{8} - 11913 x^{7} + 9115 x^{6} - 4910 x^{5} + 1756 x^{4} - 423 x^{3} + 74 x^{2} - 8 x + 1 \)

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:  \(97531030166573056000000000000=2^{24}\cdot 5^{12}\cdot 47^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.15$
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:  $2, 5, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{20} a^{16} + \frac{3}{20} a^{15} - \frac{1}{5} a^{14} + \frac{1}{5} a^{13} + \frac{7}{20} a^{12} + \frac{3}{20} a^{10} + \frac{9}{20} a^{9} - \frac{9}{20} a^{8} + \frac{1}{20} a^{7} + \frac{2}{5} a^{6} + \frac{1}{4} a^{5} - \frac{2}{5} a^{4} - \frac{9}{20} a^{3} - \frac{9}{20} a^{2} + \frac{1}{10} a + \frac{1}{20}$, $\frac{1}{20} a^{17} - \frac{3}{20} a^{15} - \frac{1}{5} a^{14} + \frac{1}{4} a^{13} + \frac{9}{20} a^{12} - \frac{7}{20} a^{11} + \frac{1}{5} a^{9} - \frac{1}{10} a^{8} + \frac{1}{4} a^{7} + \frac{1}{20} a^{6} - \frac{3}{20} a^{5} + \frac{1}{4} a^{4} + \frac{2}{5} a^{3} - \frac{1}{20} a^{2} + \frac{1}{4} a - \frac{3}{20}$, $\frac{1}{40} a^{18} - \frac{1}{8} a^{15} + \frac{3}{40} a^{14} - \frac{9}{40} a^{13} - \frac{3}{20} a^{12} - \frac{1}{2} a^{11} - \frac{17}{40} a^{10} + \frac{1}{8} a^{9} - \frac{3}{10} a^{8} - \frac{3}{20} a^{7} + \frac{1}{40} a^{6} - \frac{1}{2} a^{5} + \frac{7}{20} a^{4} - \frac{1}{5} a^{3} + \frac{9}{20} a^{2} + \frac{3}{40} a + \frac{13}{40}$, $\frac{1}{56939143701371670520} a^{19} + \frac{111473757873397373}{56939143701371670520} a^{18} + \frac{1407881241124158}{83734034854958339} a^{17} + \frac{1260474588400668823}{56939143701371670520} a^{16} + \frac{3747388757883915341}{28469571850685835260} a^{15} - \frac{1528827898432214361}{28469571850685835260} a^{14} + \frac{11557109275484368969}{56939143701371670520} a^{13} - \frac{2567469870100895161}{28469571850685835260} a^{12} - \frac{2534653247839761617}{56939143701371670520} a^{11} - \frac{554171332375786253}{14234785925342917630} a^{10} + \frac{4363670700784680021}{11387828740274334104} a^{9} - \frac{3449240846585104497}{28469571850685835260} a^{8} - \frac{20182983752156641209}{56939143701371670520} a^{7} - \frac{6098781024826298823}{56939143701371670520} a^{6} + \frac{10128801026637170587}{28469571850685835260} a^{5} + \frac{2479616613740132633}{5693914370137167052} a^{4} - \frac{7588271691893597659}{28469571850685835260} a^{3} + \frac{438593822668929257}{11387828740274334104} a^{2} - \frac{2464045258843868974}{7117392962671458815} a - \frac{7608894245384978683}{56939143701371670520}$

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:  \( 2709557.62795 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.2.19518724000000.1

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

Sibling fields

Degree 20 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 R $16{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.8.0.1}{8} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ R $16{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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
$2$2.8.16.38$x^{8} + 2 x^{6} + 4 x^{5} + 2 x^{4} + 4 x^{2} + 12$$4$$2$$16$$C_8:C_2$$[2, 3, 3]^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
$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}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
5.6.3.1$x^{6} - 10 x^{4} + 25 x^{2} - 500$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$47$47.8.0.1$x^{8} - x + 20$$1$$8$$0$$C_8$$[\ ]^{8}$
47.12.8.2$x^{12} - 103823 x^{3} + 190307559$$3$$4$$8$$C_3\times (C_3 : C_4)$$[\ ]_{3}^{12}$