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Properties

Label 20.0.47330370277...0000.2
Degree $20$
Signature $[0, 10]$
Discriminant $2^{55}\cdot 3^{16}\cdot 5^{15}$
Root discriminant $54.17$
Ramified primes $2, 3, 5$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $C_4\times F_5$ (as 20T20)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![160, 0, 1600, 0, 4176, 0, 4992, 0, 7056, 0, 5856, 0, 1704, 0, -672, 0, 114, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 + 114*x^16 - 672*x^14 + 1704*x^12 + 5856*x^10 + 7056*x^8 + 4992*x^6 + 4176*x^4 + 1600*x^2 + 160)
 
gp: K = bnfinit(x^20 + 4*x^18 + 114*x^16 - 672*x^14 + 1704*x^12 + 5856*x^10 + 7056*x^8 + 4992*x^6 + 4176*x^4 + 1600*x^2 + 160, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} + 114 x^{16} - 672 x^{14} + 1704 x^{12} + 5856 x^{10} + 7056 x^{8} + 4992 x^{6} + 4176 x^{4} + 1600 x^{2} + 160 \)

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:  \(47330370277129322496000000000000000=2^{55}\cdot 3^{16}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.17$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{8} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{8} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{8} a^{6} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} + \frac{1}{4} a^{3}$, $\frac{1}{64} a^{8} - \frac{1}{16} a^{4} - \frac{1}{2} a^{2} + \frac{5}{16}$, $\frac{1}{64} a^{9} - \frac{1}{16} a^{5} - \frac{1}{2} a^{3} + \frac{5}{16} a$, $\frac{1}{192} a^{10} - \frac{1}{192} a^{8} + \frac{1}{48} a^{6} + \frac{1}{48} a^{4} + \frac{17}{48} a^{2} - \frac{5}{48}$, $\frac{1}{192} a^{11} - \frac{1}{192} a^{9} + \frac{1}{48} a^{7} + \frac{1}{48} a^{5} + \frac{17}{48} a^{3} - \frac{5}{48} a$, $\frac{1}{1536} a^{12} - \frac{1}{384} a^{10} - \frac{1}{768} a^{8} - \frac{5}{96} a^{6} - \frac{13}{384} a^{4} + \frac{31}{96} a^{2} + \frac{7}{64}$, $\frac{1}{1536} a^{13} - \frac{1}{384} a^{11} - \frac{1}{768} a^{9} - \frac{5}{96} a^{7} - \frac{13}{384} a^{5} + \frac{31}{96} a^{3} + \frac{7}{64} a$, $\frac{1}{1536} a^{14} - \frac{1}{768} a^{10} - \frac{1}{192} a^{8} + \frac{19}{384} a^{6} - \frac{1}{48} a^{4} - \frac{25}{64} a^{2} + \frac{23}{48}$, $\frac{1}{1536} a^{15} - \frac{1}{768} a^{11} - \frac{1}{192} a^{9} + \frac{19}{384} a^{7} - \frac{1}{48} a^{5} - \frac{25}{64} a^{3} + \frac{23}{48} a$, $\frac{1}{36864} a^{16} + \frac{1}{4608} a^{14} - \frac{1}{4608} a^{12} - \frac{1}{2304} a^{10} + \frac{23}{4608} a^{8} + \frac{67}{1152} a^{6} + \frac{11}{1152} a^{4} - \frac{283}{576} a^{2} + \frac{697}{2304}$, $\frac{1}{36864} a^{17} + \frac{1}{4608} a^{15} - \frac{1}{4608} a^{13} - \frac{1}{2304} a^{11} + \frac{23}{4608} a^{9} + \frac{67}{1152} a^{7} + \frac{11}{1152} a^{5} - \frac{283}{576} a^{3} + \frac{697}{2304} a$, $\frac{1}{36864} a^{18} - \frac{1}{1536} a^{10} + \frac{1}{192} a^{6} - \frac{1}{24} a^{4} - \frac{75}{256} a^{2} - \frac{11}{36}$, $\frac{1}{36864} a^{19} - \frac{1}{1536} a^{11} + \frac{1}{192} a^{7} - \frac{1}{24} a^{5} - \frac{75}{256} a^{3} - \frac{11}{36} a$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 583394137.9828007 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_4\times F_5$ (as 20T20):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 80
The 20 conjugacy class representatives for $C_4\times F_5$
Character table for $C_4\times F_5$

Intermediate fields

\(\Q(\sqrt{10}) \), 4.0.256000.4, 5.1.648000.1, 10.2.16796160000000.1

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 R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{5}$ $20$ ${\href{/LocalNumberField/31.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ $20$

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
$3$3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
3.10.8.1$x^{10} - 3 x^{5} + 18$$5$$2$$8$$F_5$$[\ ]_{5}^{4}$
5Data not computed