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Properties

Label 20.0.32923153726...0000.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{28}\cdot 5^{12}\cdot 3469^{5}$
Root discriminant $53.20$
Ramified primes $2, 5, 3469$
Class number $1680$ (GRH)
Class group $[2, 840]$ (GRH)
Galois group 20T756

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![86725, 0, 810375, 0, 1676815, 0, 1640170, 0, 924049, 0, 324649, 0, 73265, 0, 10586, 0, 943, 0, 47, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 47*x^18 + 943*x^16 + 10586*x^14 + 73265*x^12 + 324649*x^10 + 924049*x^8 + 1640170*x^6 + 1676815*x^4 + 810375*x^2 + 86725)
 
gp: K = bnfinit(x^20 + 47*x^18 + 943*x^16 + 10586*x^14 + 73265*x^12 + 324649*x^10 + 924049*x^8 + 1640170*x^6 + 1676815*x^4 + 810375*x^2 + 86725, 1)
 

Normalized defining polynomial

\( x^{20} + 47 x^{18} + 943 x^{16} + 10586 x^{14} + 73265 x^{12} + 324649 x^{10} + 924049 x^{8} + 1640170 x^{6} + 1676815 x^{4} + 810375 x^{2} + 86725 \)

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:  \(32923153726536964440064000000000000=2^{28}\cdot 5^{12}\cdot 3469^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $53.20$
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:  $2, 5, 3469$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{10} a^{14} - \frac{1}{5} a^{12} + \frac{1}{10} a^{10} - \frac{3}{10} a^{8} - \frac{3}{10} a^{6} + \frac{1}{10} a^{4} - \frac{1}{2}$, $\frac{1}{10} a^{15} - \frac{1}{5} a^{13} + \frac{1}{10} a^{11} - \frac{3}{10} a^{9} - \frac{3}{10} a^{7} + \frac{1}{10} a^{5} - \frac{1}{2} a$, $\frac{1}{10} a^{16} + \frac{1}{5} a^{12} - \frac{1}{10} a^{10} - \frac{1}{2} a^{9} - \frac{2}{5} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{3}{10} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{10} a^{17} + \frac{1}{5} a^{13} - \frac{1}{10} a^{11} + \frac{1}{10} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{5} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{836968543300} a^{18} - \frac{1}{20} a^{17} + \frac{1539667147}{836968543300} a^{16} + \frac{6215928979}{418484271650} a^{14} - \frac{1}{10} a^{13} - \frac{24888404441}{209242135825} a^{12} - \frac{1}{5} a^{11} - \frac{17339966551}{167393708660} a^{10} - \frac{1}{20} a^{9} - \frac{266879164391}{836968543300} a^{8} + \frac{15613990366}{209242135825} a^{6} + \frac{2}{5} a^{5} - \frac{4582847953}{83696854330} a^{4} + \frac{80377747539}{167393708660} a^{2} + \frac{1}{4} a + \frac{13541150937}{33478741732}$, $\frac{1}{836968543300} a^{19} - \frac{20154380009}{418484271650} a^{17} - \frac{1}{20} a^{16} + \frac{6215928979}{418484271650} a^{15} - \frac{91625236047}{418484271650} a^{13} - \frac{1}{10} a^{12} + \frac{32878146047}{167393708660} a^{11} - \frac{1}{5} a^{10} + \frac{54878340047}{418484271650} a^{9} + \frac{9}{20} a^{8} - \frac{178014155093}{418484271650} a^{7} - \frac{6476266693}{41848427165} a^{5} - \frac{1}{10} a^{4} - \frac{3319106791}{167393708660} a^{3} + \frac{1292866376}{8369685433} a - \frac{1}{4}$

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

Class group and class number

$C_{2}\times C_{840}$, which has order $1680$ (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:  \( 828338.933858 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T756:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 102400
The 130 conjugacy class representatives for t20n756 are not computed
Character table for t20n756 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.10.9627168800000.1

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 R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/29.10.0.1}{10} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
3469Data not computed