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Properties

Label 20.20.3172126672...2672.1
Degree $20$
Signature $[20, 0]$
Discriminant $2^{20}\cdot 11^{16}\cdot 23^{3}\cdot 419^{6}$
Root discriminant $133.37$
Ramified primes $2, 11, 23, 419$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T432

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![708898288583, 0, -896697119210, 0, 453844496710, 0, -118717836375, 0, 17521759023, 0, -1521153905, 0, 78516261, 0, -2362410, 0, 39182, 0, -321, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 321*x^18 + 39182*x^16 - 2362410*x^14 + 78516261*x^12 - 1521153905*x^10 + 17521759023*x^8 - 118717836375*x^6 + 453844496710*x^4 - 896697119210*x^2 + 708898288583)
 
gp: K = bnfinit(x^20 - 321*x^18 + 39182*x^16 - 2362410*x^14 + 78516261*x^12 - 1521153905*x^10 + 17521759023*x^8 - 118717836375*x^6 + 453844496710*x^4 - 896697119210*x^2 + 708898288583, 1)
 

Normalized defining polynomial

\( x^{20} - 321 x^{18} + 39182 x^{16} - 2362410 x^{14} + 78516261 x^{12} - 1521153905 x^{10} + 17521759023 x^{8} - 118717836375 x^{6} + 453844496710 x^{4} - 896697119210 x^{2} + 708898288583 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[20, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(3172126672449549143567477325812349950492672=2^{20}\cdot 11^{16}\cdot 23^{3}\cdot 419^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.37$
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:  $2, 11, 23, 419$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{419} a^{14} + \frac{98}{419} a^{12} - \frac{204}{419} a^{10} - \frac{88}{419} a^{8} - \frac{149}{419} a^{6} + \frac{36}{419} a^{4} - \frac{156}{419} a^{2}$, $\frac{1}{419} a^{15} + \frac{98}{419} a^{13} - \frac{204}{419} a^{11} - \frac{88}{419} a^{9} - \frac{149}{419} a^{7} + \frac{36}{419} a^{5} - \frac{156}{419} a^{3}$, $\frac{1}{1046870243} a^{16} + \frac{941172}{1046870243} a^{14} - \frac{366610958}{1046870243} a^{12} + \frac{242368786}{1046870243} a^{10} - \frac{19784491}{1046870243} a^{8} - \frac{180052644}{1046870243} a^{6} + \frac{191089822}{1046870243} a^{4} - \frac{416260}{2498497} a^{2} + \frac{1768}{5963}$, $\frac{1}{1046870243} a^{17} + \frac{941172}{1046870243} a^{15} - \frac{366610958}{1046870243} a^{13} + \frac{242368786}{1046870243} a^{11} - \frac{19784491}{1046870243} a^{9} - \frac{180052644}{1046870243} a^{7} + \frac{191089822}{1046870243} a^{5} - \frac{416260}{2498497} a^{3} + \frac{1768}{5963} a$, $\frac{1}{313014454295924169430101461798986302208283} a^{18} - \frac{11634484477511937712691064127834}{313014454295924169430101461798986302208283} a^{16} - \frac{105045803681668424050065311073738067394}{313014454295924169430101461798986302208283} a^{14} + \frac{118572701172891077145170536177427349127270}{313014454295924169430101461798986302208283} a^{12} - \frac{39258485026461601202788738208960081108962}{313014454295924169430101461798986302208283} a^{10} + \frac{59385215104206442569404994707370931189435}{313014454295924169430101461798986302208283} a^{8} + \frac{134395892051022064154375803251793251328138}{313014454295924169430101461798986302208283} a^{6} - \frac{310502909317518503905726280590296250371}{747051203570224748043201579472521007657} a^{4} + \frac{29893763213751716882861023382635698}{1782938433341825174327450070340145603} a^{2} + \frac{1056676610789594599869306575593957}{4255222991269272492428281790787937}$, $\frac{1}{313014454295924169430101461798986302208283} a^{19} - \frac{11634484477511937712691064127834}{313014454295924169430101461798986302208283} a^{17} - \frac{105045803681668424050065311073738067394}{313014454295924169430101461798986302208283} a^{15} + \frac{118572701172891077145170536177427349127270}{313014454295924169430101461798986302208283} a^{13} - \frac{39258485026461601202788738208960081108962}{313014454295924169430101461798986302208283} a^{11} + \frac{59385215104206442569404994707370931189435}{313014454295924169430101461798986302208283} a^{9} + \frac{134395892051022064154375803251793251328138}{313014454295924169430101461798986302208283} a^{7} - \frac{310502909317518503905726280590296250371}{747051203570224748043201579472521007657} a^{5} + \frac{29893763213751716882861023382635698}{1782938433341825174327450070340145603} a^{3} + \frac{1056676610789594599869306575593957}{4255222991269272492428281790787937} a$

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:  $19$
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:  \( 827787980943000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T432:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 10240
The 136 conjugacy class representatives for t20n432 are not computed
Character table for t20n432 is not computed

Intermediate fields

\(\Q(\zeta_{11})^+\), 10.10.2065776536197.1

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 ${\href{/LocalNumberField/3.10.0.1}{10} }{,}\,{\href{/LocalNumberField/3.5.0.1}{5} }^{2}$ $20$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ R ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/19.5.0.1}{5} }^{4}$ R ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}$ $20$ $20$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.10.0.1}{10} }{,}\,{\href{/LocalNumberField/47.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}$

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
$11$11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
11.5.4.4$x^{5} - 11$$5$$1$$4$$C_5$$[\ ]_{5}$
$23$23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
23.4.3.1$x^{4} + 46$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
419Data not computed