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Properties

Label 20.0.12976980681...6944.1
Degree $20$
Signature $[0, 10]$
Discriminant $2^{38}\cdot 31^{2}\cdot 53^{12}$
Root discriminant $56.97$
Ramified primes $2, 31, 53$
Class number $180$ (GRH)
Class group $[3, 60]$ (GRH)
Galois group 20T196

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1048576, 0, 262144, 0, 114688, 0, 4096, 0, 1280, 0, -896, 0, 80, 0, 16, 0, 28, 0, 4, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 4*x^18 + 28*x^16 + 16*x^14 + 80*x^12 - 896*x^10 + 1280*x^8 + 4096*x^6 + 114688*x^4 + 262144*x^2 + 1048576)
 
gp: K = bnfinit(x^20 + 4*x^18 + 28*x^16 + 16*x^14 + 80*x^12 - 896*x^10 + 1280*x^8 + 4096*x^6 + 114688*x^4 + 262144*x^2 + 1048576, 1)
 

Normalized defining polynomial

\( x^{20} + 4 x^{18} + 28 x^{16} + 16 x^{14} + 80 x^{12} - 896 x^{10} + 1280 x^{8} + 4096 x^{6} + 114688 x^{4} + 262144 x^{2} + 1048576 \)

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:  \(129769806814273798102061261409746944=2^{38}\cdot 31^{2}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.97$
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:  $2, 31, 53$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{16} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{7} - \frac{1}{16} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{64} a^{8} - \frac{1}{32} a^{6} + \frac{1}{16} a^{4}$, $\frac{1}{128} a^{9} - \frac{1}{64} a^{7} + \frac{1}{32} a^{5} - \frac{1}{4} a^{3}$, $\frac{1}{128} a^{10} + \frac{1}{16} a^{4}$, $\frac{1}{256} a^{11} + \frac{1}{32} a^{5} - \frac{1}{2} a$, $\frac{1}{512} a^{12} - \frac{1}{128} a^{8} + \frac{1}{32} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{1024} a^{13} - \frac{1}{256} a^{9} + \frac{1}{64} a^{5} - \frac{1}{16} a^{3}$, $\frac{1}{4096} a^{14} - \frac{1}{1024} a^{10} - \frac{1}{128} a^{8} - \frac{1}{64} a^{7} - \frac{3}{256} a^{6} + \frac{1}{32} a^{5} + \frac{1}{64} a^{4} - \frac{1}{16} a^{3} + \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{8192} a^{15} - \frac{1}{2048} a^{11} - \frac{1}{256} a^{9} - \frac{3}{512} a^{7} - \frac{1}{32} a^{6} + \frac{1}{128} a^{5} + \frac{1}{16} a^{4} + \frac{1}{16} a^{3} - \frac{1}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{65536} a^{16} - \frac{1}{16384} a^{14} + \frac{15}{16384} a^{12} - \frac{13}{4096} a^{10} - \frac{19}{4096} a^{8} - \frac{1}{64} a^{7} + \frac{1}{128} a^{6} + \frac{1}{32} a^{5} + \frac{21}{256} a^{4} - \frac{1}{16} a^{3} - \frac{1}{32} a^{2} - \frac{1}{2} a + \frac{1}{4}$, $\frac{1}{262144} a^{17} + \frac{3}{65536} a^{15} + \frac{15}{65536} a^{13} - \frac{17}{16384} a^{11} + \frac{13}{16384} a^{9} + \frac{7}{1024} a^{7} - \frac{55}{1024} a^{5} + \frac{27}{128} a^{3} + \frac{5}{16} a$, $\frac{1}{1572864} a^{18} - \frac{1}{131072} a^{16} + \frac{7}{393216} a^{14} + \frac{7}{32768} a^{12} - \frac{283}{98304} a^{10} - \frac{89}{12288} a^{8} - \frac{5}{2048} a^{6} + \frac{5}{48} a^{4} + \frac{5}{12}$, $\frac{1}{6291456} a^{19} - \frac{1}{524288} a^{17} + \frac{7}{1572864} a^{15} - \frac{57}{131072} a^{13} + \frac{485}{393216} a^{11} - \frac{185}{49152} a^{9} + \frac{59}{8192} a^{7} + \frac{7}{384} a^{5} + \frac{1}{32} a^{3} - \frac{7}{48} a$

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

Class group and class number

$C_{3}\times C_{60}$, which has order $180$ (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:  \( 63506693.7844 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T196:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1280
The 44 conjugacy class representatives for t20n196
Character table for t20n196 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 5.5.2382032.1, 10.10.11620508567601152.1, 10.0.1407170959357952.1, 10.0.45029470699454464.1

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ R ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}$

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$2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.2.3.1$x^{2} + 14$$2$$1$$3$$C_2$$[3]$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
2.8.16.3$x^{8} + 2 x^{6} + 6 x^{4} + 4 x^{2} + 8 x + 28$$4$$2$$16$$C_4\times C_2$$[2, 3]^{2}$
31Data not computed
$53$53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.0.1$x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
53.8.6.1$x^{8} - 1643 x^{4} + 1755625$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$