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Properties

Label 20.4.20623838732...2736.1
Degree $20$
Signature $[4, 8]$
Discriminant $2^{42}\cdot 97^{2}\cdot 2657^{4}$
Root discriminant $32.79$
Ramified primes $2, 97, 2657$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1028

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -2, -32, 1, -48, 312, 48, 766, 32, 948, 32, 766, 48, 312, -48, 1, -32, -2, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 2*x^18 - 32*x^17 + x^16 - 48*x^15 + 312*x^14 + 48*x^13 + 766*x^12 + 32*x^11 + 948*x^10 + 32*x^9 + 766*x^8 + 48*x^7 + 312*x^6 - 48*x^5 + x^4 - 32*x^3 - 2*x^2 + 1)
 
gp: K = bnfinit(x^20 - 2*x^18 - 32*x^17 + x^16 - 48*x^15 + 312*x^14 + 48*x^13 + 766*x^12 + 32*x^11 + 948*x^10 + 32*x^9 + 766*x^8 + 48*x^7 + 312*x^6 - 48*x^5 + x^4 - 32*x^3 - 2*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{20} - 2 x^{18} - 32 x^{17} + x^{16} - 48 x^{15} + 312 x^{14} + 48 x^{13} + 766 x^{12} + 32 x^{11} + 948 x^{10} + 32 x^{9} + 766 x^{8} + 48 x^{7} + 312 x^{6} - 48 x^{5} + x^{4} - 32 x^{3} - 2 x^{2} + 1 \)

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:  \(2062383873207852920714335092736=2^{42}\cdot 97^{2}\cdot 2657^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.79$
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:  $2, 97, 2657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{4} a^{9} - \frac{1}{4} a$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{8} - \frac{1}{8} a^{4} + \frac{1}{8}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{10} - \frac{1}{8} a^{6} + \frac{1}{8} a^{2}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{3}{16} a^{7} - \frac{3}{16} a^{6} - \frac{3}{16} a^{5} - \frac{3}{16} a^{4} + \frac{3}{16} a^{3} - \frac{3}{16} a^{2} - \frac{3}{16} a - \frac{3}{16}$, $\frac{1}{16} a^{16} - \frac{1}{8} a^{8} + \frac{1}{16}$, $\frac{1}{16} a^{17} - \frac{1}{8} a^{9} + \frac{1}{16} a$, $\frac{1}{692896} a^{18} + \frac{8515}{346448} a^{17} + \frac{615}{692896} a^{16} + \frac{9821}{346448} a^{15} - \frac{9051}{173224} a^{14} + \frac{8509}{346448} a^{13} - \frac{9327}{173224} a^{12} - \frac{36595}{346448} a^{11} - \frac{347}{5872} a^{10} + \frac{38027}{346448} a^{9} + \frac{387}{5872} a^{8} + \frac{6711}{346448} a^{7} + \frac{33979}{173224} a^{6} + \frac{51815}{346448} a^{5} + \frac{34255}{173224} a^{4} + \frac{53127}{346448} a^{3} + \frac{43921}{692896} a^{2} + \frac{80043}{173224} a - \frac{43305}{692896}$, $\frac{1}{692896} a^{19} - \frac{3}{692896} a^{17} - \frac{8531}{346448} a^{16} - \frac{307}{346448} a^{15} + \frac{738}{21653} a^{14} - \frac{3395}{346448} a^{13} + \frac{823}{21653} a^{12} + \frac{20345}{173224} a^{11} - \frac{7087}{86612} a^{10} - \frac{353}{173224} a^{9} - \frac{8179}{173224} a^{8} + \frac{85815}{346448} a^{7} + \frac{3642}{21653} a^{6} + \frac{40463}{346448} a^{5} + \frac{820}{21653} a^{4} - \frac{7101}{692896} a^{3} + \frac{24607}{86612} a^{2} + \frac{172607}{692896} a - \frac{95127}{346448}$

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:  $11$
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:  \( 37942201.7817 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1028:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 228 conjugacy class representatives for t20n1028 are not computed
Character table for t20n1028 is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), 10.6.925322313728.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: 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 ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/7.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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$2.8.16.16$x^{8} + 2 x^{6} + 4 x^{5} + 6 x^{4} + 8 x^{3} + 4$$4$$2$$16$$D_4\times C_2$$[2, 2, 3]^{2}$
2.12.26.87$x^{12} + 4 x^{11} + 2 x^{10} + 2 x^{8} + 2 x^{6} + 2 x^{4} + 4 x^{3} + 4 x^{2} + 2$$12$$1$$26$12T48$[4/3, 4/3, 2, 3]_{3}^{2}$
97Data not computed
2657Data not computed