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Properties

Label 20.0.88330463788...8832.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{27}\cdot 7^{10}\cdot 13^{12}$
Root discriminant $31.43$
Ramified primes $2, 7, 13$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $D_4\times F_5$ (as 20T42)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![32768, 0, 4096, 0, 512, 0, -2112, 0, 440, 0, 59, 0, 55, 0, -33, 0, 1, 0, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + x^18 + x^16 - 33*x^14 + 55*x^12 + 59*x^10 + 440*x^8 - 2112*x^6 + 512*x^4 + 4096*x^2 + 32768)
 
gp: K = bnfinit(x^20 + x^18 + x^16 - 33*x^14 + 55*x^12 + 59*x^10 + 440*x^8 - 2112*x^6 + 512*x^4 + 4096*x^2 + 32768, 1)
 

Normalized defining polynomial

\( x^{20} + x^{18} + x^{16} - 33 x^{14} + 55 x^{12} + 59 x^{10} + 440 x^{8} - 2112 x^{6} + 512 x^{4} + 4096 x^{2} + 32768 \)

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:  \(883304637884625106516765048832=2^{27}\cdot 7^{10}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.43$
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:  $2, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} - \frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{16} a^{13} + \frac{1}{16} a^{11} + \frac{1}{16} a^{9} - \frac{1}{16} a^{7} + \frac{7}{16} a^{5} - \frac{5}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{64} a^{14} + \frac{1}{64} a^{12} + \frac{1}{64} a^{10} - \frac{1}{64} a^{8} - \frac{9}{64} a^{6} - \frac{5}{64} a^{4} + \frac{3}{8} a^{2}$, $\frac{1}{128} a^{15} + \frac{1}{128} a^{13} + \frac{1}{128} a^{11} - \frac{1}{128} a^{9} - \frac{9}{128} a^{7} - \frac{5}{128} a^{5} - \frac{1}{2} a^{4} - \frac{5}{16} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2048} a^{16} - \frac{7}{2048} a^{14} - \frac{1}{32} a^{13} + \frac{121}{2048} a^{12} - \frac{1}{32} a^{11} + \frac{87}{2048} a^{10} - \frac{1}{32} a^{9} - \frac{65}{2048} a^{8} + \frac{1}{32} a^{7} + \frac{3}{2048} a^{6} + \frac{9}{32} a^{5} + \frac{27}{64} a^{4} + \frac{5}{32} a^{3} - \frac{5}{16} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{8192} a^{17} - \frac{7}{8192} a^{15} - \frac{1}{128} a^{14} - \frac{135}{8192} a^{13} - \frac{1}{128} a^{12} + \frac{343}{8192} a^{11} - \frac{1}{128} a^{10} + \frac{703}{8192} a^{9} + \frac{1}{128} a^{8} - \frac{1789}{8192} a^{7} + \frac{9}{128} a^{6} + \frac{19}{256} a^{5} - \frac{59}{128} a^{4} + \frac{21}{64} a^{3} + \frac{5}{16} a^{2} + \frac{3}{16} a$, $\frac{1}{3915776} a^{18} - \frac{279}{3915776} a^{16} - \frac{1}{256} a^{15} - \frac{28951}{3915776} a^{14} - \frac{1}{256} a^{13} + \frac{45255}{3915776} a^{12} - \frac{1}{256} a^{11} - \frac{174513}{3915776} a^{10} - \frac{31}{256} a^{9} + \frac{390675}{3915776} a^{8} + \frac{9}{256} a^{7} + \frac{22515}{244736} a^{6} - \frac{59}{256} a^{5} + \frac{4841}{30592} a^{4} - \frac{15}{32} a^{3} - \frac{3313}{7648} a^{2} - \frac{99}{478}$, $\frac{1}{15663104} a^{19} - \frac{279}{15663104} a^{17} - \frac{28951}{15663104} a^{15} - \frac{1}{128} a^{14} + \frac{45255}{15663104} a^{13} + \frac{7}{128} a^{12} + \frac{804431}{15663104} a^{11} + \frac{7}{128} a^{10} - \frac{588269}{15663104} a^{9} + \frac{9}{128} a^{8} - \frac{222221}{978944} a^{7} + \frac{1}{128} a^{6} - \frac{18103}{122368} a^{5} - \frac{3}{128} a^{4} - \frac{5225}{30592} a^{3} - \frac{1}{2} a^{2} - \frac{99}{1912} a - \frac{1}{2}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 7649961.18437 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_4\times F_5$ (as 20T42):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 160
The 25 conjugacy class representatives for $D_4\times F_5$
Character table for $D_4\times F_5$ is not computed

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.392.1, 5.5.6889792.1, 10.0.332284636622848.1

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 $20$ ${\href{/LocalNumberField/5.4.0.1}{4} }^{5}$ R ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{5}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }{,}\,{\href{/LocalNumberField/43.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{5}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2}$
2.8.12.1$x^{8} + 6 x^{6} + 8 x^{5} + 16$$2$$4$$12$$C_4\times C_2$$[3]^{4}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
7.8.4.1$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed