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Properties

Label 20.4.54014487747...1600.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{42}\cdot 5^{2}\cdot 53^{12}$
Root discriminant $54.53$
Ramified primes $2, 5, 53$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group 20T196

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5776, 0, -14602, 0, -4023, 0, 6404, 0, 16096, 0, 12288, 0, 5878, 0, 1676, 0, 288, 0, 26, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 + 26*x^18 + 288*x^16 + 1676*x^14 + 5878*x^12 + 12288*x^10 + 16096*x^8 + 6404*x^6 - 4023*x^4 - 14602*x^2 + 5776)
 
gp: K = bnfinit(x^20 + 26*x^18 + 288*x^16 + 1676*x^14 + 5878*x^12 + 12288*x^10 + 16096*x^8 + 6404*x^6 - 4023*x^4 - 14602*x^2 + 5776, 1)
 

Normalized defining polynomial

\( x^{20} + 26 x^{18} + 288 x^{16} + 1676 x^{14} + 5878 x^{12} + 12288 x^{10} + 16096 x^{8} + 6404 x^{6} - 4023 x^{4} - 14602 x^{2} + 5776 \)

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

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[4, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(54014487747876710968599900690841600=2^{42}\cdot 5^{2}\cdot 53^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.53$
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:  $2, 5, 53$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{6} a^{6} + \frac{1}{3} a^{4} + \frac{1}{6} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{7} + \frac{1}{3} a^{5} + \frac{1}{6} a^{3} + \frac{1}{3} a$, $\frac{1}{6} a^{8} - \frac{1}{2} a^{4} + \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} + \frac{1}{3} a$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{2}$, $\frac{1}{12} a^{11} - \frac{1}{2} a^{5} - \frac{1}{12} a^{3} - \frac{1}{2} a$, $\frac{1}{36} a^{12} - \frac{1}{18} a^{10} + \frac{1}{18} a^{6} + \frac{5}{12} a^{4} + \frac{1}{9} a^{2} + \frac{4}{9}$, $\frac{1}{72} a^{13} - \frac{1}{72} a^{12} - \frac{1}{36} a^{11} + \frac{1}{36} a^{10} - \frac{1}{18} a^{7} + \frac{1}{18} a^{6} - \frac{11}{24} a^{5} + \frac{11}{24} a^{4} + \frac{17}{36} a^{3} - \frac{17}{36} a^{2} - \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{72} a^{14} - \frac{1}{72} a^{12} - \frac{1}{36} a^{10} - \frac{1}{18} a^{8} - \frac{1}{72} a^{6} + \frac{1}{72} a^{4} - \frac{17}{36} a^{2} - \frac{4}{9}$, $\frac{1}{72} a^{15} - \frac{1}{72} a^{12} + \frac{1}{36} a^{11} + \frac{1}{36} a^{10} - \frac{1}{18} a^{9} - \frac{5}{72} a^{7} + \frac{1}{18} a^{6} + \frac{1}{18} a^{5} + \frac{11}{24} a^{4} - \frac{1}{12} a^{3} - \frac{17}{36} a^{2} - \frac{7}{18} a + \frac{4}{9}$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{12} + \frac{1}{36} a^{10} - \frac{5}{72} a^{8} + \frac{1}{18} a^{6} - \frac{1}{24} a^{4} + \frac{1}{36} a^{2}$, $\frac{1}{144} a^{17} - \frac{1}{144} a^{16} - \frac{1}{144} a^{13} + \frac{1}{144} a^{12} + \frac{1}{72} a^{11} + \frac{5}{72} a^{10} + \frac{7}{144} a^{9} - \frac{7}{144} a^{8} + \frac{1}{36} a^{7} - \frac{1}{36} a^{6} - \frac{13}{48} a^{5} - \frac{11}{48} a^{4} - \frac{35}{72} a^{3} + \frac{29}{72} a^{2} + \frac{1}{6} a + \frac{1}{3}$, $\frac{1}{1641848939568} a^{18} - \frac{1101300593}{182427659952} a^{16} + \frac{754067509}{182427659952} a^{14} - \frac{6626290135}{1641848939568} a^{12} - \frac{31210462417}{547282979856} a^{10} - \frac{326642821}{14791431888} a^{8} - \frac{50753011961}{1641848939568} a^{6} + \frac{66017708681}{547282979856} a^{4} + \frac{15168667845}{30404609992} a^{2} + \frac{32615179814}{102615558723}$, $\frac{1}{31195129851792} a^{19} - \frac{1776119507}{2599594154316} a^{17} - \frac{1}{144} a^{16} + \frac{32666812519}{10398376617264} a^{15} - \frac{31817466935}{15597564925896} a^{13} - \frac{1}{144} a^{12} + \frac{68141755007}{3466125539088} a^{11} - \frac{5}{72} a^{10} - \frac{1157891615}{35129650734} a^{9} - \frac{7}{144} a^{8} + \frac{815778372811}{31195129851792} a^{7} + \frac{1}{36} a^{6} - \frac{472303661551}{1733062769544} a^{5} + \frac{11}{48} a^{4} - \frac{182940573778}{649898538579} a^{3} + \frac{7}{72} a^{2} + \frac{625505074658}{1949695615737} a - \frac{2}{9}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $11$
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:  \( 10138718168.6 \) (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, 10.2.58102542838005760.1, 10.10.11620508567601152.1, 10.2.453926115921920.2

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 ${\href{/LocalNumberField/3.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{10}$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ 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
2Data not computed
$5$5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
5.8.0.1$x^{8} + x^{2} - 2 x + 3$$1$$8$$0$$C_8$$[\ ]^{8}$
$53$53.2.0.1$x^{2} - x + 5$$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}$